Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 490341 17 pageshttpdxdoiorg1011552013490341

Research ArticleThe Traveling Wave Solutions and Their Bifurcations forthe BBM-Like 119861(119898 119899) Equations

Shaoyong Li12 and Zhengrong Liu2

1 College of Mathematics and Information Sciences Shaoguan University Shaoguan Guangdong 512005 China2Department of Mathematics South China University of Technology Guangzhou Guangdong 510640 China

Correspondence should be addressed to Shaoyong Li lishaoyongoksohucom

Received 3 June 2013 Accepted 3 July 2013

Academic Editor Shiping Lu

Copyright copy 2013 S Li and Z Liu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate the travelingwave solutions and their bifurcations for the BBM-like119861(119898 119899) equations 119906119905+120572119906119909+120573(119906119898)119909minus120574(119906119899)119909119909119905= 0

by using bifurcation method and numerical simulation approach of dynamical systems Firstly for BBM-like B(3 2) equation weobtain some precise expressions of traveling wave solutions which include periodic blow-up and periodic wave solution peakonand periodic peakon wave solution and solitary wave and blow-up solution Furthermore we reveal the relationships amongthese solutions theoretically Secondly for BBM-like B(4 2) equation we construct two periodic wave solutions and two blow-upsolutions In order to confirm the correctness of these solutions we also check them by software Mathematica

1 Introduction

In recent years the nonlinear phenomena exist in all fieldsincluding either the scientific work or engineering fieldssuch as fluid mechanics plasma physics optical fibers biol-ogy solid-state physics chemical kinematics and chemicalphysics Many nonlinear evolution equations are playingimportant roles in the analysis of the phenomena

In order to find the traveling wave solutions of these non-linear evolution equations there have been many methodssuch as Jacobi elliptic function method [1 2] F-expansionand extended F-expansion method [3 4] (1198661015840119866)-expansionmethod [5 6] and the bifurcation method of dynamicalsystems [7ndash11]

BBM equation or regularized long-wave equation (RLWequation)

119906119905+ 119906119906119909minus 119906119909119909119905= 0 (1)

was derived by Peregine [12 13] and Benjamin et al [14] as analternative model to Korteweg-de Vries equation for small-amplitude long wavelength surface water waves

There are various generalized form related to (1) Shang[15] introduced a family of BBM-like equations with nonlin-ear dispersion

119906119905+ (119906119898)119909minus (119906119899)119909119909119905= 0 119898 119899 gt 1 (2)

which were called BBM-like 119861(119898 119899) equations as alternativemodel to the nonlinear dispersive119870(119898 119899) equations [16ndash18]He presented a method called the extend sine-cosine methodto seek exact solitary-wave solutions with compact supportand exact special solutions with solitary patterns of (2)

When 119898 = 119899 = 2 (2) reduces to the BBM-like 119861(2 2)equation

119906119905+ (1199062)119909minus (1199062)119909119909119905= 0 (3)

Jiang et al [19] employed the bifurcation method of dynam-ical systems to investigate (3) Under different parametricconditions they gave various sufficient conditions to guaran-tee the existence of smooth and nonsmooth traveling wavesolutions Furthermore through some special phase orbitsthey obtained some solitary wave solutions expressed byimplicit functions periodic cusp wave solution compactonsolution and peakon solution

2 Journal of Applied Mathematics

Wazwaz [20] introduced a system of nonlinear variantRLW equations

119906119905+ 119886119906119909minus 119896(119906119899)119909+ 119887(119906119899)119909119909119905= 0 (4)

and derived some compact and noncompact exact solutionsby using the sine-cosine method and tanh method

Feng et al [21] studied the following generalized variantRLW equations

119906119905+ 119886119906119909minus 119896(119906119898)119909+ 119887(119906119899)119909119909119905= 0 (5)

By using four different ansatzs they obtained some exactsolutions such as compactons solitary pattern solutionssolitons and periodic solutions

Kuru [22ndash24] considered the following BBM-like equa-tions with a fully nonlinear dispersive term

119906119905+ 119906119909+ 119886(119906119898)119909minus (119906119899)119909119909119905= 0 119898 119899 gt 1 (6)

By means of the factorization technique he obtained thetraveling wave solutions of (6) in terms of the Weierstrassfunctions

In the present paper we use the bifurcation methodand numerical simulation approach of dynamical systems tostudy the following BBM-like 119861(119898 119899) equations

119906119905+ 120572119906119909+ 120573(119906

119898)119909minus 120574(119906119899)119909119909119905= 0 119898 119899 gt 1 (7)

For BBM-like 119861(3 2) equation we obtain some preciseexpressions of traveling wave solutions which include peri-odic blow-up and periodic wave solution peakon and peri-odic peakon wave solution and solitary wave and blow-up solution We also reveal the relationships among thesesolutions theoretically For BBM-like 119861(4 2) equation weconstruct two elliptic periodic wave solutions and two hyper-bolic blow-up solutions

This paper is organized as follows In Section 2 we stateour main results which are included in two propositions InSections 3 and 4 we give the derivations for the two proposi-tions respectively A brief conclusion is given in Section 5

2 Main Results and Remarks

In this section we list our main results and give someremarks Firstly let us recall some symbolsThe symbols sn 119906and cn 119906 denote the Jacobian elliptic functions sine amplitude119906 and cosine amplitude 119906 cosh 119906 sinh 119906 sech 119906 and csch119906 are the hyperbolic functions Secondly for the sake ofsimplification we only consider the case 120572 gt 0 120573 gt 0 and120574 gt 0 (the other cases can be considered similarly) To relateconveniently for given constant wave speed 119888 let

120585 = 119909 minus 119888119905

1198920=4

27

radic5|119888 minus 120572|

3

120573

(8)

Proposition 1 Consider BBM-like 119861(3 2) equation

119906119905+ 120572119906119909+ 120573(119906

3)119909minus 120574(119906

2)119909119909119905= 0 (9)

and its traveling wave equation

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (10)

For given constants 119888 and 119892 there are the followingresults

(1)When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus1198920 (9) has two elliptic

periodic blow-up solutions

1199061(120585) = 120593

1+ 1198601minus

21198601

1 minus cn (1205781120585 1198961)

1199062(120585) = 120593

1+ 1198601minus

21198601

1 + cn (1205781120585 1198961)

(11)

where

1198601= radic(120593

1minus 1205932) (1205931minus 1205933) (12)

1205781= radic

1205731198601

5120574119888 (13)

1198961= radic

21198601+ 21205931minus 1205932minus 1205933

41198601

(14)

1205931=23radic100120573 (120572 minus 119888) minus

3radic10Ω

23

6120573Ω13 (15)

1205932=

3radic5 (2120573

3radic10 (1 minus radic3119894) (119888 minus 120572) + (1 + radic3119894)Ω

23)

63radic4120573Ω13

(16)

1205933=

3radic5 (2120573

3radic10 (1 + radic3119894) (119888 minus 120572) + (1 minus radic3119894)Ω

23)

63radic4120573Ω13

(17)

Ω = radic72911989221205734 minus 80(119888 minus 120572)31205733 minus 27119892120573

2 (18)

For the graphs of 1199061(120585) and 119906

2(120585) see Figures 1(a) and 1(b)

(2)When 119888 gt 120572 and 119892 = minus1198920 (9) has two trigonometric

periodic blow-up solutions

1199063(120585) = radic

5 (119888 minus 120572)

9120573(1 minus 3csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)) (19)

1199064(120585) = radic

5 (119888 minus 120572)

9120573(1 minus 3sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)) (20)

The graphs of 1199063(120585) and 119906

4(120585) are similar to Figure 1

Journal of Applied Mathematics 3

minus30 minus20 minus10minus10

minus20

minus30

minus40

minus50

minus60

10 20 30

(a) 119906 = 1199061(120585)

minus30 minus20 minus10

minus20

minus40

minus80

minus60

10 20 30

(b) 119906 = 1199062(120585)

Figure 1 The graphs of 1199061(120585) and 119906

2(120585) when 120572 = 120573 = 120574 = 1 119888 = 2 and 119892 = minus2radic39

(3)When 119888 gt 120572 and minus1198920lt 119892 lt 119892

0 (9) has two elliptic

periodic blow-up solutions 1199065(120585) 1199066(120585) and two symmetric

elliptic periodic wave solutions 1199067(120585) 1199068(120585)

1199065(120585) = 120593

3minus (1205933minus 1205931) snminus2 (120578

2120585 1198962)

1199066(120585) =

1205931minus 1205932sn2 (120578

2120585 1198962)

1 minus sn2 (1205782120585 1198962)

1199067(120585) = 120593

3minus (1205933minus 1205932) sn2 (120578

2120585 1198962)

1199068(120585) =

1205932minus 12059311198962

2sn2 (120578

2120585 1198962)

1 minus 1198962

2sn2 (120578

2120585 1198962)

(21)

where

1205782=1

2

radic120573 (1205933minus 1205931)

5120574119888

1198962= radic

1205933minus 1205932

1205933minus 1205931

(22)

For the graphs of 119906119894(120585) (119894 = 5ndash8) see Figures 2ndash4

(4)When 119888 gt 120572 and 119892 = 1198920 (9) has a hyperbolic smooth

solitary wave solution

1199069(120585) = minusradic

5 (119888 minus 120572)

9120573(1 minus 3sech2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585))

(23)

a hyperbolic blow-up solution

11990610(120585) = minusradic

5 (119888 minus 120572)

9120573(1 + 3csch2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585))

(24)

a hyperbolic peakon wave solution

11990611(120585) = minusradic

5 (119888 minus 120572)

9120573

times (1 minus 3sech2(1205751minus radic

120573

120574119888

4radic119888 minus 120572

80120573

10038161003816100381610038161205851003816100381610038161003816))

(25)

and a hyperbolic periodic peakon wave solution

11990612(120585) = minusradic

5 (119888 minus 120572)

9120573

times (1 minus 3sech2(1205751+ radic

120573

120574119888

4radic119888 minus 120572

80120573

10038161003816100381610038161205851003816100381610038161003816))

120585 isin [(2119899 minus 1) 119879 (2119899 + 1) 119879)

(26)

where

1205751=1

2ln (5 minus 2radic6)

119879 = radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

119899 = 0 plusmn1 plusmn2

(27)

For the graphs of 1199069(120585) and 119906

10(120585) see Figures 4(d) and 3(d)

For the graphs of 11990611(120585) and 119906

12(120585) see Figures 7(a) and 7(b)

(5)When 119888 gt 120572 and 119892 gt 1198920 (9) has two elliptic periodic

blow-up solutions

11990613(120585) = 120593

3+ 1198602minus

21198602

1 minus cn (1205783120585 1198963)

11990614(120585) = 120593

3+ 1198602minus

21198602

1 + cn (1205783120585 1198963)

(28)

where

1198602= radic(120593

3minus 1205932) (1205933minus 1205931)

1205783= radic

1205731198602

5120574119888

1198963= radic

21198602+ 21205933minus 1205932minus 1205931

41198602

(29)

For the graphs of 11990613(120585) and 119906

14(120585) see Figures 5(a) and 6(a)

4 Journal of Applied Mathematics

12

1

08

06

04

02

minus20 minus10 0 10 20

u7 (120585)

u8 (120585)

(a)

12

1

08

06

04

02

minus20 minus10 0 10 20

(b)

12

1

08

06

04

02

minus20 minus10 0 10 20

(c)

12

1

08

06

04

02

minus20 minus10 0 10 20

(d)

Figure 2The varying process for graphs of 1199067(120585) and 119906

8(120585)when 119888 gt 120572 and 119892 rarr minus119892

0+0where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = minus119892

0+10minus1

(b) 119892 = minus1198920+ 10minus2 (c) 119892 = minus119892

0+ 10minus4 (d) 119892 = minus119892

0+ 10minus6

(6) When 119888 lt 0 (9) has two elliptic periodic blow-upsolutions

11990615(120585) = 120593

1minus 1198601+

21198601

1 minus cn (1205784120585 1198964)

11990616(120585) = 120593

1minus 1198601+

21198601

1 + cn (1205784120585 1198964)

(30)

1205784= radic

1205731198601

minus5120574119888

1198964= radic

21198601minus 21205931+ 1205932+ 1205933

41198601

(31)

For the graphs of 11990615(120585) and 119906

16(120585) see Figures 7(c) and 7(d)

Remark 2 When 119888 gt 120572 and119892 rarr minus1198920minus0 the elliptic periodic

blow-up solutions 1199061(120585) and 119906

2(120585) become the trigonometric

periodic blow-up solutions 1199063(120585) and 119906

4(120585) respectively

Remark 3 When 119888 gt 120572 and119892 rarr minus1198920+0 the elliptic periodic

blow-up solutions 1199065(120585) and 119906

6(120585) become the trigonometric

periodic blow-up solutions 1199063(120585) and 119906

4(120585) respectively

The symmetric elliptic periodic wave solutions 1199067(120585) and

1199068(120585) become a trivial solution 119906(120585) = radic5(119888 minus 120572)9120573 and for

the varying process see Figure 2

Remark 4 When 119888 gt 120572 and 119892 rarr 1198920minus 0 the elliptic periodic

blow-up solution 1199065(120585) becomes the hyperbolic blow-up

solution 11990610(120585) for the varying process see Figure 3 The

elliptic periodic wave solutions 1199067(120585) become the hyperbolic

smooth solitary wave solution 1199069(120585) and for the varying

process see Figure 4 The elliptic solutions 1199066(120585) and 119906

8(120585)

become a trivial solution 119906(120585) = minusradic5(119888 minus 120572)9120573

Remark 5 When 119888 gt 120572 and 119892 rarr 1198920+ 0 the elliptic periodic

blow-up solution 11990613(120585) becomes the hyperbolic blow-up

solution 11990610(120585) and for the varying process see Figure 5

The elliptic periodic blow-up solution 11990614(120585) becomes the

hyperbolic smooth solitary wave solution 1199069(120585) and for the

varying process see Figure 6

Proposition 6 Consider BBM-like 119861(4 2) equation

119906119905+ 120572119906119909+ 120573(119906

4)119909minus 120574(119906

2)119909119909119905= 0 (32)

Journal of Applied Mathematics 5

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(a)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(b)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(c)

minus10 10 20 30

minus20

minus40

minus20minus30

minus60

minus80

(d)

Figure 3 The varying process for graphs of 1199065(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

1

05

minus20 minus10 10 20

(a)

1

15

05

minus05

minus20 minus10 10 20

(b)

1

15

05

minus05

minus20 minus10 10 20

(c)

1

15

05

minus05

minus20 minus10 10 20

(d)

Figure 4 The varying process for graphs of 1199067(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

6 Journal of Applied Mathematics

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(b)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(c)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(d)

Figure 5 The varying process for graphs of 11990613(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus5

minus10

minus15

minus20

(b)

minus40 minus20 20 40

2

minus8

minus10

minus6

minus4

minus2

(c)

15

1

05

minus30 minus20 minus10 10 20 30

minus05

minus1

(d)

Figure 6 The varying process for graphs of 11990614(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

Journal of Applied Mathematics 7

minus10minus15 minus5 5 10 15minus01

minus02

minus03

minus04

minus05

minus06

minus07

(a) 119906 = 11990611(120585)

14

12

1

08

06

04

02

minus40 minus20 20 40

(b) 119906 = 11990612(120585)

60

50

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990615(120585)

50

40

30

20

10

minus20 minus10 10 20

(d) 119906 = 11990616(120585)

Figure 7 The graphs of 11990611(120585) 11990612(120585) when 119888 = 2 119892 = 4radic527 and 119906

15(120585) 11990616(120585) when 119888 = minus1 119892 = minus10

and its traveling wave equation

(120572 minus 119888) 120595 + 1205731205954+ 2120574119888(120595

1015840)2

+ 212057411988812059512059510158401015840= 119892 (33)

For given constants 119888 and 119892 there are the followingresults

(1∘) When 119888 gt 120572 and 119892 = 0 (32) has two elliptic periodicwave solutions

11990617(120585) =

1 minus cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 + (radic3 minus 1) cn (120578

5120585 1198965))

11990618(120585) =

1 + cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 minus (radic3 minus 1) cn (120578

5120585 1198965))

(34)

where

1205785=4radic3radic

119888 minus 120572

3120574119888

6radic

120573

2 (119888 minus 120572)

1198965=radic3 minus 1

2radic2

(35)

(2∘) When 119888 lt 0 and 119892 = minus 3radic(119888 minus 120572)416120573 (32) has twohyperbolic blow-up solutions

11990619(120585) =

3radic119888 minus 120572 (minus4 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (2 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

11990620(120585) =

3radic119888 minus 120572 (4 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (minus2 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

(36)

where

1205786= minusradic

120573

minus120574119888

3radic119888 minus 120572

2120573 (37)

For the graphs of 119906119894(120585) (119894 = 17ndash20) see Figure 8

Remark 7 In order to confirm the correctness of thesesolutions we have verified them by using the softwareMathematica for instance about 119906

20(120585) the commands are as

follows

120585 = 119909 minus 119888119905

120578 = minusradic120573

minus120574119888

3radic119888 minus 120572

3radic2120573

8 Journal of Applied Mathematics

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(a) 119906 = 11990617(120585)

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(b) 119906 = 11990618(120585)

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990619(120585)

minus40

minus30

minus20

minus10

minus20 minus10 10 20

(d) 119906 = 11990620(120585)

Figure 8 The graphs of 11990617(120585) 11990618(120585) when 119888 = 20 119892 = 0 and 119906

19(120585) 11990620(120585) when 119888 = minus16

119906 =

3radic119888 minus 120572 (4 + 2 cosh [120578120585] + radic6 sinh [120578120585])3radic2120573 (minus2 + 2 cosh [120578120585] + radic6 sinh [120578120585])

Simplify [119863 [119906 119905] + 120572119863 [119906 119909]

+ 120573119863 [1199064 119909] minus 120574119863 [119906

2 119909 2 119905]]

0

(38)

3 The Derivations for Proposition 1

In this section firstly we derive the precise expressions ofthe traveling wave solutions for BBM-like 119861(3 2) equationSecondly we show the relationships among these solutionstheoretically Substituting 119906(119909 119905) = 120593(120585) with 120585 = 119909 minus 119888119905 into(9) it follows that

(120572 minus 119888) 1205931015840+ 120573(120593

3)1015840

+ 120574119888(1205932)101584010158401015840

= 0 (39)

Integrating (39) once we have

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (40)

where 119892 is an integral constant

Letting 119910 = 1205931015840 we obtain the following planar system

119889120593

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

2120574119888120593

(41)

Under the transformation

119889120585 = 2120574119888120593119889120591 (42)

system (41) becomes

119889120593

119889120591= 2120574119888120593119910

119889119910

119889120591= 119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

(43)

Clearly system (41) and system (43) have the same firstintegral

12057411988812059321199102minus119892

21205932minus119888 minus 120572

31205933+120573

51205935= ℎ (44)

where ℎ is an integral constant Consequently these twosystems have the same topological phase portraits except for

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

Wazwaz [20] introduced a system of nonlinear variantRLW equations

119906119905+ 119886119906119909minus 119896(119906119899)119909+ 119887(119906119899)119909119909119905= 0 (4)

and derived some compact and noncompact exact solutionsby using the sine-cosine method and tanh method

Feng et al [21] studied the following generalized variantRLW equations

119906119905+ 119886119906119909minus 119896(119906119898)119909+ 119887(119906119899)119909119909119905= 0 (5)

By using four different ansatzs they obtained some exactsolutions such as compactons solitary pattern solutionssolitons and periodic solutions

Kuru [22ndash24] considered the following BBM-like equa-tions with a fully nonlinear dispersive term

119906119905+ 119906119909+ 119886(119906119898)119909minus (119906119899)119909119909119905= 0 119898 119899 gt 1 (6)

By means of the factorization technique he obtained thetraveling wave solutions of (6) in terms of the Weierstrassfunctions

In the present paper we use the bifurcation methodand numerical simulation approach of dynamical systems tostudy the following BBM-like 119861(119898 119899) equations

119906119905+ 120572119906119909+ 120573(119906

119898)119909minus 120574(119906119899)119909119909119905= 0 119898 119899 gt 1 (7)

For BBM-like 119861(3 2) equation we obtain some preciseexpressions of traveling wave solutions which include peri-odic blow-up and periodic wave solution peakon and peri-odic peakon wave solution and solitary wave and blow-up solution We also reveal the relationships among thesesolutions theoretically For BBM-like 119861(4 2) equation weconstruct two elliptic periodic wave solutions and two hyper-bolic blow-up solutions

This paper is organized as follows In Section 2 we stateour main results which are included in two propositions InSections 3 and 4 we give the derivations for the two proposi-tions respectively A brief conclusion is given in Section 5

2 Main Results and Remarks

In this section we list our main results and give someremarks Firstly let us recall some symbolsThe symbols sn 119906and cn 119906 denote the Jacobian elliptic functions sine amplitude119906 and cosine amplitude 119906 cosh 119906 sinh 119906 sech 119906 and csch119906 are the hyperbolic functions Secondly for the sake ofsimplification we only consider the case 120572 gt 0 120573 gt 0 and120574 gt 0 (the other cases can be considered similarly) To relateconveniently for given constant wave speed 119888 let

120585 = 119909 minus 119888119905

1198920=4

27

radic5|119888 minus 120572|

3

120573

(8)

Proposition 1 Consider BBM-like 119861(3 2) equation

119906119905+ 120572119906119909+ 120573(119906

3)119909minus 120574(119906

2)119909119909119905= 0 (9)

and its traveling wave equation

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (10)

For given constants 119888 and 119892 there are the followingresults

(1)When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus1198920 (9) has two elliptic

periodic blow-up solutions

1199061(120585) = 120593

1+ 1198601minus

21198601

1 minus cn (1205781120585 1198961)

1199062(120585) = 120593

1+ 1198601minus

21198601

1 + cn (1205781120585 1198961)

(11)

where

1198601= radic(120593

1minus 1205932) (1205931minus 1205933) (12)

1205781= radic

1205731198601

5120574119888 (13)

1198961= radic

21198601+ 21205931minus 1205932minus 1205933

41198601

(14)

1205931=23radic100120573 (120572 minus 119888) minus

3radic10Ω

23

6120573Ω13 (15)

1205932=

3radic5 (2120573

3radic10 (1 minus radic3119894) (119888 minus 120572) + (1 + radic3119894)Ω

23)

63radic4120573Ω13

(16)

1205933=

3radic5 (2120573

3radic10 (1 + radic3119894) (119888 minus 120572) + (1 minus radic3119894)Ω

23)

63radic4120573Ω13

(17)

Ω = radic72911989221205734 minus 80(119888 minus 120572)31205733 minus 27119892120573

2 (18)

For the graphs of 1199061(120585) and 119906

2(120585) see Figures 1(a) and 1(b)

(2)When 119888 gt 120572 and 119892 = minus1198920 (9) has two trigonometric

periodic blow-up solutions

1199063(120585) = radic

5 (119888 minus 120572)

9120573(1 minus 3csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)) (19)

1199064(120585) = radic

5 (119888 minus 120572)

9120573(1 minus 3sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)) (20)

The graphs of 1199063(120585) and 119906

4(120585) are similar to Figure 1

Journal of Applied Mathematics 3

minus30 minus20 minus10minus10

minus20

minus30

minus40

minus50

minus60

10 20 30

(a) 119906 = 1199061(120585)

minus30 minus20 minus10

minus20

minus40

minus80

minus60

10 20 30

(b) 119906 = 1199062(120585)

Figure 1 The graphs of 1199061(120585) and 119906

2(120585) when 120572 = 120573 = 120574 = 1 119888 = 2 and 119892 = minus2radic39

(3)When 119888 gt 120572 and minus1198920lt 119892 lt 119892

0 (9) has two elliptic

periodic blow-up solutions 1199065(120585) 1199066(120585) and two symmetric

elliptic periodic wave solutions 1199067(120585) 1199068(120585)

1199065(120585) = 120593

3minus (1205933minus 1205931) snminus2 (120578

2120585 1198962)

1199066(120585) =

1205931minus 1205932sn2 (120578

2120585 1198962)

1 minus sn2 (1205782120585 1198962)

1199067(120585) = 120593

3minus (1205933minus 1205932) sn2 (120578

2120585 1198962)

1199068(120585) =

1205932minus 12059311198962

2sn2 (120578

2120585 1198962)

1 minus 1198962

2sn2 (120578

2120585 1198962)

(21)

where

1205782=1

2

radic120573 (1205933minus 1205931)

5120574119888

1198962= radic

1205933minus 1205932

1205933minus 1205931

(22)

For the graphs of 119906119894(120585) (119894 = 5ndash8) see Figures 2ndash4

(4)When 119888 gt 120572 and 119892 = 1198920 (9) has a hyperbolic smooth

solitary wave solution

1199069(120585) = minusradic

5 (119888 minus 120572)

9120573(1 minus 3sech2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585))

(23)

a hyperbolic blow-up solution

11990610(120585) = minusradic

5 (119888 minus 120572)

9120573(1 + 3csch2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585))

(24)

a hyperbolic peakon wave solution

11990611(120585) = minusradic

5 (119888 minus 120572)

9120573

times (1 minus 3sech2(1205751minus radic

120573

120574119888

4radic119888 minus 120572

80120573

10038161003816100381610038161205851003816100381610038161003816))

(25)

and a hyperbolic periodic peakon wave solution

11990612(120585) = minusradic

5 (119888 minus 120572)

9120573

times (1 minus 3sech2(1205751+ radic

120573

120574119888

4radic119888 minus 120572

80120573

10038161003816100381610038161205851003816100381610038161003816))

120585 isin [(2119899 minus 1) 119879 (2119899 + 1) 119879)

(26)

where

1205751=1

2ln (5 minus 2radic6)

119879 = radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

119899 = 0 plusmn1 plusmn2

(27)

For the graphs of 1199069(120585) and 119906

10(120585) see Figures 4(d) and 3(d)

For the graphs of 11990611(120585) and 119906

12(120585) see Figures 7(a) and 7(b)

(5)When 119888 gt 120572 and 119892 gt 1198920 (9) has two elliptic periodic

blow-up solutions

11990613(120585) = 120593

3+ 1198602minus

21198602

1 minus cn (1205783120585 1198963)

11990614(120585) = 120593

3+ 1198602minus

21198602

1 + cn (1205783120585 1198963)

(28)

where

1198602= radic(120593

3minus 1205932) (1205933minus 1205931)

1205783= radic

1205731198602

5120574119888

1198963= radic

21198602+ 21205933minus 1205932minus 1205931

41198602

(29)

For the graphs of 11990613(120585) and 119906

14(120585) see Figures 5(a) and 6(a)

4 Journal of Applied Mathematics

12

1

08

06

04

02

minus20 minus10 0 10 20

u7 (120585)

u8 (120585)

(a)

12

1

08

06

04

02

minus20 minus10 0 10 20

(b)

12

1

08

06

04

02

minus20 minus10 0 10 20

(c)

12

1

08

06

04

02

minus20 minus10 0 10 20

(d)

Figure 2The varying process for graphs of 1199067(120585) and 119906

8(120585)when 119888 gt 120572 and 119892 rarr minus119892

0+0where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = minus119892

0+10minus1

(b) 119892 = minus1198920+ 10minus2 (c) 119892 = minus119892

0+ 10minus4 (d) 119892 = minus119892

0+ 10minus6

(6) When 119888 lt 0 (9) has two elliptic periodic blow-upsolutions

11990615(120585) = 120593

1minus 1198601+

21198601

1 minus cn (1205784120585 1198964)

11990616(120585) = 120593

1minus 1198601+

21198601

1 + cn (1205784120585 1198964)

(30)

1205784= radic

1205731198601

minus5120574119888

1198964= radic

21198601minus 21205931+ 1205932+ 1205933

41198601

(31)

For the graphs of 11990615(120585) and 119906

16(120585) see Figures 7(c) and 7(d)

Remark 2 When 119888 gt 120572 and119892 rarr minus1198920minus0 the elliptic periodic

blow-up solutions 1199061(120585) and 119906

2(120585) become the trigonometric

periodic blow-up solutions 1199063(120585) and 119906

4(120585) respectively

Remark 3 When 119888 gt 120572 and119892 rarr minus1198920+0 the elliptic periodic

blow-up solutions 1199065(120585) and 119906

6(120585) become the trigonometric

periodic blow-up solutions 1199063(120585) and 119906

4(120585) respectively

The symmetric elliptic periodic wave solutions 1199067(120585) and

1199068(120585) become a trivial solution 119906(120585) = radic5(119888 minus 120572)9120573 and for

the varying process see Figure 2

Remark 4 When 119888 gt 120572 and 119892 rarr 1198920minus 0 the elliptic periodic

blow-up solution 1199065(120585) becomes the hyperbolic blow-up

solution 11990610(120585) for the varying process see Figure 3 The

elliptic periodic wave solutions 1199067(120585) become the hyperbolic

smooth solitary wave solution 1199069(120585) and for the varying

process see Figure 4 The elliptic solutions 1199066(120585) and 119906

8(120585)

become a trivial solution 119906(120585) = minusradic5(119888 minus 120572)9120573

Remark 5 When 119888 gt 120572 and 119892 rarr 1198920+ 0 the elliptic periodic

blow-up solution 11990613(120585) becomes the hyperbolic blow-up

solution 11990610(120585) and for the varying process see Figure 5

The elliptic periodic blow-up solution 11990614(120585) becomes the

hyperbolic smooth solitary wave solution 1199069(120585) and for the

varying process see Figure 6

Proposition 6 Consider BBM-like 119861(4 2) equation

119906119905+ 120572119906119909+ 120573(119906

4)119909minus 120574(119906

2)119909119909119905= 0 (32)

Journal of Applied Mathematics 5

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(a)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(b)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(c)

minus10 10 20 30

minus20

minus40

minus20minus30

minus60

minus80

(d)

Figure 3 The varying process for graphs of 1199065(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

1

05

minus20 minus10 10 20

(a)

1

15

05

minus05

minus20 minus10 10 20

(b)

1

15

05

minus05

minus20 minus10 10 20

(c)

1

15

05

minus05

minus20 minus10 10 20

(d)

Figure 4 The varying process for graphs of 1199067(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

6 Journal of Applied Mathematics

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(b)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(c)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(d)

Figure 5 The varying process for graphs of 11990613(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus5

minus10

minus15

minus20

(b)

minus40 minus20 20 40

2

minus8

minus10

minus6

minus4

minus2

(c)

15

1

05

minus30 minus20 minus10 10 20 30

minus05

minus1

(d)

Figure 6 The varying process for graphs of 11990614(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

Journal of Applied Mathematics 7

minus10minus15 minus5 5 10 15minus01

minus02

minus03

minus04

minus05

minus06

minus07

(a) 119906 = 11990611(120585)

14

12

1

08

06

04

02

minus40 minus20 20 40

(b) 119906 = 11990612(120585)

60

50

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990615(120585)

50

40

30

20

10

minus20 minus10 10 20

(d) 119906 = 11990616(120585)

Figure 7 The graphs of 11990611(120585) 11990612(120585) when 119888 = 2 119892 = 4radic527 and 119906

15(120585) 11990616(120585) when 119888 = minus1 119892 = minus10

and its traveling wave equation

(120572 minus 119888) 120595 + 1205731205954+ 2120574119888(120595

1015840)2

+ 212057411988812059512059510158401015840= 119892 (33)

For given constants 119888 and 119892 there are the followingresults

(1∘) When 119888 gt 120572 and 119892 = 0 (32) has two elliptic periodicwave solutions

11990617(120585) =

1 minus cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 + (radic3 minus 1) cn (120578

5120585 1198965))

11990618(120585) =

1 + cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 minus (radic3 minus 1) cn (120578

5120585 1198965))

(34)

where

1205785=4radic3radic

119888 minus 120572

3120574119888

6radic

120573

2 (119888 minus 120572)

1198965=radic3 minus 1

2radic2

(35)

(2∘) When 119888 lt 0 and 119892 = minus 3radic(119888 minus 120572)416120573 (32) has twohyperbolic blow-up solutions

11990619(120585) =

3radic119888 minus 120572 (minus4 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (2 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

11990620(120585) =

3radic119888 minus 120572 (4 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (minus2 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

(36)

where

1205786= minusradic

120573

minus120574119888

3radic119888 minus 120572

2120573 (37)

For the graphs of 119906119894(120585) (119894 = 17ndash20) see Figure 8

Remark 7 In order to confirm the correctness of thesesolutions we have verified them by using the softwareMathematica for instance about 119906

20(120585) the commands are as

follows

120585 = 119909 minus 119888119905

120578 = minusradic120573

minus120574119888

3radic119888 minus 120572

3radic2120573

8 Journal of Applied Mathematics

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(a) 119906 = 11990617(120585)

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(b) 119906 = 11990618(120585)

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990619(120585)

minus40

minus30

minus20

minus10

minus20 minus10 10 20

(d) 119906 = 11990620(120585)

Figure 8 The graphs of 11990617(120585) 11990618(120585) when 119888 = 20 119892 = 0 and 119906

19(120585) 11990620(120585) when 119888 = minus16

119906 =

3radic119888 minus 120572 (4 + 2 cosh [120578120585] + radic6 sinh [120578120585])3radic2120573 (minus2 + 2 cosh [120578120585] + radic6 sinh [120578120585])

Simplify [119863 [119906 119905] + 120572119863 [119906 119909]

+ 120573119863 [1199064 119909] minus 120574119863 [119906

2 119909 2 119905]]

0

(38)

3 The Derivations for Proposition 1

In this section firstly we derive the precise expressions ofthe traveling wave solutions for BBM-like 119861(3 2) equationSecondly we show the relationships among these solutionstheoretically Substituting 119906(119909 119905) = 120593(120585) with 120585 = 119909 minus 119888119905 into(9) it follows that

(120572 minus 119888) 1205931015840+ 120573(120593

3)1015840

+ 120574119888(1205932)101584010158401015840

= 0 (39)

Integrating (39) once we have

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (40)

where 119892 is an integral constant

Letting 119910 = 1205931015840 we obtain the following planar system

119889120593

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

2120574119888120593

(41)

Under the transformation

119889120585 = 2120574119888120593119889120591 (42)

system (41) becomes

119889120593

119889120591= 2120574119888120593119910

119889119910

119889120591= 119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

(43)

Clearly system (41) and system (43) have the same firstintegral

12057411988812059321199102minus119892

21205932minus119888 minus 120572

31205933+120573

51205935= ℎ (44)

where ℎ is an integral constant Consequently these twosystems have the same topological phase portraits except for

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

minus30 minus20 minus10minus10

minus20

minus30

minus40

minus50

minus60

10 20 30

(a) 119906 = 1199061(120585)

minus30 minus20 minus10

minus20

minus40

minus80

minus60

10 20 30

(b) 119906 = 1199062(120585)

Figure 1 The graphs of 1199061(120585) and 119906

2(120585) when 120572 = 120573 = 120574 = 1 119888 = 2 and 119892 = minus2radic39

(3)When 119888 gt 120572 and minus1198920lt 119892 lt 119892

0 (9) has two elliptic

periodic blow-up solutions 1199065(120585) 1199066(120585) and two symmetric

elliptic periodic wave solutions 1199067(120585) 1199068(120585)

1199065(120585) = 120593

3minus (1205933minus 1205931) snminus2 (120578

2120585 1198962)

1199066(120585) =

1205931minus 1205932sn2 (120578

2120585 1198962)

1 minus sn2 (1205782120585 1198962)

1199067(120585) = 120593

3minus (1205933minus 1205932) sn2 (120578

2120585 1198962)

1199068(120585) =

1205932minus 12059311198962

2sn2 (120578

2120585 1198962)

1 minus 1198962

2sn2 (120578

2120585 1198962)

(21)

where

1205782=1

2

radic120573 (1205933minus 1205931)

5120574119888

1198962= radic

1205933minus 1205932

1205933minus 1205931

(22)

For the graphs of 119906119894(120585) (119894 = 5ndash8) see Figures 2ndash4

(4)When 119888 gt 120572 and 119892 = 1198920 (9) has a hyperbolic smooth

solitary wave solution

1199069(120585) = minusradic

5 (119888 minus 120572)

9120573(1 minus 3sech2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585))

(23)

a hyperbolic blow-up solution

11990610(120585) = minusradic

5 (119888 minus 120572)

9120573(1 + 3csch2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585))

(24)

a hyperbolic peakon wave solution

11990611(120585) = minusradic

5 (119888 minus 120572)

9120573

times (1 minus 3sech2(1205751minus radic

120573

120574119888

4radic119888 minus 120572

80120573

10038161003816100381610038161205851003816100381610038161003816))

(25)

and a hyperbolic periodic peakon wave solution

11990612(120585) = minusradic

5 (119888 minus 120572)

9120573

times (1 minus 3sech2(1205751+ radic

120573

120574119888

4radic119888 minus 120572

80120573

10038161003816100381610038161205851003816100381610038161003816))

120585 isin [(2119899 minus 1) 119879 (2119899 + 1) 119879)

(26)

where

1205751=1

2ln (5 minus 2radic6)

119879 = radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

119899 = 0 plusmn1 plusmn2

(27)

For the graphs of 1199069(120585) and 119906

10(120585) see Figures 4(d) and 3(d)

For the graphs of 11990611(120585) and 119906

12(120585) see Figures 7(a) and 7(b)

(5)When 119888 gt 120572 and 119892 gt 1198920 (9) has two elliptic periodic

blow-up solutions

11990613(120585) = 120593

3+ 1198602minus

21198602

1 minus cn (1205783120585 1198963)

11990614(120585) = 120593

3+ 1198602minus

21198602

1 + cn (1205783120585 1198963)

(28)

where

1198602= radic(120593

3minus 1205932) (1205933minus 1205931)

1205783= radic

1205731198602

5120574119888

1198963= radic

21198602+ 21205933minus 1205932minus 1205931

41198602

(29)

For the graphs of 11990613(120585) and 119906

14(120585) see Figures 5(a) and 6(a)

4 Journal of Applied Mathematics

12

1

08

06

04

02

minus20 minus10 0 10 20

u7 (120585)

u8 (120585)

(a)

12

1

08

06

04

02

minus20 minus10 0 10 20

(b)

12

1

08

06

04

02

minus20 minus10 0 10 20

(c)

12

1

08

06

04

02

minus20 minus10 0 10 20

(d)

Figure 2The varying process for graphs of 1199067(120585) and 119906

8(120585)when 119888 gt 120572 and 119892 rarr minus119892

0+0where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = minus119892

0+10minus1

(b) 119892 = minus1198920+ 10minus2 (c) 119892 = minus119892

0+ 10minus4 (d) 119892 = minus119892

0+ 10minus6

(6) When 119888 lt 0 (9) has two elliptic periodic blow-upsolutions

11990615(120585) = 120593

1minus 1198601+

21198601

1 minus cn (1205784120585 1198964)

11990616(120585) = 120593

1minus 1198601+

21198601

1 + cn (1205784120585 1198964)

(30)

1205784= radic

1205731198601

minus5120574119888

1198964= radic

21198601minus 21205931+ 1205932+ 1205933

41198601

(31)

For the graphs of 11990615(120585) and 119906

16(120585) see Figures 7(c) and 7(d)

Remark 2 When 119888 gt 120572 and119892 rarr minus1198920minus0 the elliptic periodic

blow-up solutions 1199061(120585) and 119906

2(120585) become the trigonometric

periodic blow-up solutions 1199063(120585) and 119906

4(120585) respectively

Remark 3 When 119888 gt 120572 and119892 rarr minus1198920+0 the elliptic periodic

blow-up solutions 1199065(120585) and 119906

6(120585) become the trigonometric

periodic blow-up solutions 1199063(120585) and 119906

4(120585) respectively

The symmetric elliptic periodic wave solutions 1199067(120585) and

1199068(120585) become a trivial solution 119906(120585) = radic5(119888 minus 120572)9120573 and for

the varying process see Figure 2

Remark 4 When 119888 gt 120572 and 119892 rarr 1198920minus 0 the elliptic periodic

blow-up solution 1199065(120585) becomes the hyperbolic blow-up

solution 11990610(120585) for the varying process see Figure 3 The

elliptic periodic wave solutions 1199067(120585) become the hyperbolic

smooth solitary wave solution 1199069(120585) and for the varying

process see Figure 4 The elliptic solutions 1199066(120585) and 119906

8(120585)

become a trivial solution 119906(120585) = minusradic5(119888 minus 120572)9120573

Remark 5 When 119888 gt 120572 and 119892 rarr 1198920+ 0 the elliptic periodic

blow-up solution 11990613(120585) becomes the hyperbolic blow-up

solution 11990610(120585) and for the varying process see Figure 5

The elliptic periodic blow-up solution 11990614(120585) becomes the

hyperbolic smooth solitary wave solution 1199069(120585) and for the

varying process see Figure 6

Proposition 6 Consider BBM-like 119861(4 2) equation

119906119905+ 120572119906119909+ 120573(119906

4)119909minus 120574(119906

2)119909119909119905= 0 (32)

Journal of Applied Mathematics 5

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(a)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(b)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(c)

minus10 10 20 30

minus20

minus40

minus20minus30

minus60

minus80

(d)

Figure 3 The varying process for graphs of 1199065(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

1

05

minus20 minus10 10 20

(a)

1

15

05

minus05

minus20 minus10 10 20

(b)

1

15

05

minus05

minus20 minus10 10 20

(c)

1

15

05

minus05

minus20 minus10 10 20

(d)

Figure 4 The varying process for graphs of 1199067(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

6 Journal of Applied Mathematics

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(b)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(c)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(d)

Figure 5 The varying process for graphs of 11990613(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus5

minus10

minus15

minus20

(b)

minus40 minus20 20 40

2

minus8

minus10

minus6

minus4

minus2

(c)

15

1

05

minus30 minus20 minus10 10 20 30

minus05

minus1

(d)

Figure 6 The varying process for graphs of 11990614(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

Journal of Applied Mathematics 7

minus10minus15 minus5 5 10 15minus01

minus02

minus03

minus04

minus05

minus06

minus07

(a) 119906 = 11990611(120585)

14

12

1

08

06

04

02

minus40 minus20 20 40

(b) 119906 = 11990612(120585)

60

50

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990615(120585)

50

40

30

20

10

minus20 minus10 10 20

(d) 119906 = 11990616(120585)

Figure 7 The graphs of 11990611(120585) 11990612(120585) when 119888 = 2 119892 = 4radic527 and 119906

15(120585) 11990616(120585) when 119888 = minus1 119892 = minus10

and its traveling wave equation

(120572 minus 119888) 120595 + 1205731205954+ 2120574119888(120595

1015840)2

+ 212057411988812059512059510158401015840= 119892 (33)

For given constants 119888 and 119892 there are the followingresults

(1∘) When 119888 gt 120572 and 119892 = 0 (32) has two elliptic periodicwave solutions

11990617(120585) =

1 minus cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 + (radic3 minus 1) cn (120578

5120585 1198965))

11990618(120585) =

1 + cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 minus (radic3 minus 1) cn (120578

5120585 1198965))

(34)

where

1205785=4radic3radic

119888 minus 120572

3120574119888

6radic

120573

2 (119888 minus 120572)

1198965=radic3 minus 1

2radic2

(35)

(2∘) When 119888 lt 0 and 119892 = minus 3radic(119888 minus 120572)416120573 (32) has twohyperbolic blow-up solutions

11990619(120585) =

3radic119888 minus 120572 (minus4 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (2 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

11990620(120585) =

3radic119888 minus 120572 (4 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (minus2 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

(36)

where

1205786= minusradic

120573

minus120574119888

3radic119888 minus 120572

2120573 (37)

For the graphs of 119906119894(120585) (119894 = 17ndash20) see Figure 8

Remark 7 In order to confirm the correctness of thesesolutions we have verified them by using the softwareMathematica for instance about 119906

20(120585) the commands are as

follows

120585 = 119909 minus 119888119905

120578 = minusradic120573

minus120574119888

3radic119888 minus 120572

3radic2120573

8 Journal of Applied Mathematics

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(a) 119906 = 11990617(120585)

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(b) 119906 = 11990618(120585)

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990619(120585)

minus40

minus30

minus20

minus10

minus20 minus10 10 20

(d) 119906 = 11990620(120585)

Figure 8 The graphs of 11990617(120585) 11990618(120585) when 119888 = 20 119892 = 0 and 119906

19(120585) 11990620(120585) when 119888 = minus16

119906 =

3radic119888 minus 120572 (4 + 2 cosh [120578120585] + radic6 sinh [120578120585])3radic2120573 (minus2 + 2 cosh [120578120585] + radic6 sinh [120578120585])

Simplify [119863 [119906 119905] + 120572119863 [119906 119909]

+ 120573119863 [1199064 119909] minus 120574119863 [119906

2 119909 2 119905]]

0

(38)

3 The Derivations for Proposition 1

In this section firstly we derive the precise expressions ofthe traveling wave solutions for BBM-like 119861(3 2) equationSecondly we show the relationships among these solutionstheoretically Substituting 119906(119909 119905) = 120593(120585) with 120585 = 119909 minus 119888119905 into(9) it follows that

(120572 minus 119888) 1205931015840+ 120573(120593

3)1015840

+ 120574119888(1205932)101584010158401015840

= 0 (39)

Integrating (39) once we have

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (40)

where 119892 is an integral constant

Letting 119910 = 1205931015840 we obtain the following planar system

119889120593

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

2120574119888120593

(41)

Under the transformation

119889120585 = 2120574119888120593119889120591 (42)

system (41) becomes

119889120593

119889120591= 2120574119888120593119910

119889119910

119889120591= 119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

(43)

Clearly system (41) and system (43) have the same firstintegral

12057411988812059321199102minus119892

21205932minus119888 minus 120572

31205933+120573

51205935= ℎ (44)

where ℎ is an integral constant Consequently these twosystems have the same topological phase portraits except for

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

12

1

08

06

04

02

minus20 minus10 0 10 20

u7 (120585)

u8 (120585)

(a)

12

1

08

06

04

02

minus20 minus10 0 10 20

(b)

12

1

08

06

04

02

minus20 minus10 0 10 20

(c)

12

1

08

06

04

02

minus20 minus10 0 10 20

(d)

Figure 2The varying process for graphs of 1199067(120585) and 119906

8(120585)when 119888 gt 120572 and 119892 rarr minus119892

0+0where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = minus119892

0+10minus1

(b) 119892 = minus1198920+ 10minus2 (c) 119892 = minus119892

0+ 10minus4 (d) 119892 = minus119892

0+ 10minus6

(6) When 119888 lt 0 (9) has two elliptic periodic blow-upsolutions

11990615(120585) = 120593

1minus 1198601+

21198601

1 minus cn (1205784120585 1198964)

11990616(120585) = 120593

1minus 1198601+

21198601

1 + cn (1205784120585 1198964)

(30)

1205784= radic

1205731198601

minus5120574119888

1198964= radic

21198601minus 21205931+ 1205932+ 1205933

41198601

(31)

For the graphs of 11990615(120585) and 119906

16(120585) see Figures 7(c) and 7(d)

Remark 2 When 119888 gt 120572 and119892 rarr minus1198920minus0 the elliptic periodic

blow-up solutions 1199061(120585) and 119906

2(120585) become the trigonometric

periodic blow-up solutions 1199063(120585) and 119906

4(120585) respectively

Remark 3 When 119888 gt 120572 and119892 rarr minus1198920+0 the elliptic periodic

blow-up solutions 1199065(120585) and 119906

6(120585) become the trigonometric

periodic blow-up solutions 1199063(120585) and 119906

4(120585) respectively

The symmetric elliptic periodic wave solutions 1199067(120585) and

1199068(120585) become a trivial solution 119906(120585) = radic5(119888 minus 120572)9120573 and for

the varying process see Figure 2

Remark 4 When 119888 gt 120572 and 119892 rarr 1198920minus 0 the elliptic periodic

blow-up solution 1199065(120585) becomes the hyperbolic blow-up

solution 11990610(120585) for the varying process see Figure 3 The

elliptic periodic wave solutions 1199067(120585) become the hyperbolic

smooth solitary wave solution 1199069(120585) and for the varying

process see Figure 4 The elliptic solutions 1199066(120585) and 119906

8(120585)

become a trivial solution 119906(120585) = minusradic5(119888 minus 120572)9120573

Remark 5 When 119888 gt 120572 and 119892 rarr 1198920+ 0 the elliptic periodic

blow-up solution 11990613(120585) becomes the hyperbolic blow-up

solution 11990610(120585) and for the varying process see Figure 5

The elliptic periodic blow-up solution 11990614(120585) becomes the

hyperbolic smooth solitary wave solution 1199069(120585) and for the

varying process see Figure 6

Proposition 6 Consider BBM-like 119861(4 2) equation

119906119905+ 120572119906119909+ 120573(119906

4)119909minus 120574(119906

2)119909119909119905= 0 (32)

Journal of Applied Mathematics 5

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(a)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(b)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(c)

minus10 10 20 30

minus20

minus40

minus20minus30

minus60

minus80

(d)

Figure 3 The varying process for graphs of 1199065(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

1

05

minus20 minus10 10 20

(a)

1

15

05

minus05

minus20 minus10 10 20

(b)

1

15

05

minus05

minus20 minus10 10 20

(c)

1

15

05

minus05

minus20 minus10 10 20

(d)

Figure 4 The varying process for graphs of 1199067(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

6 Journal of Applied Mathematics

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(b)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(c)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(d)

Figure 5 The varying process for graphs of 11990613(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus5

minus10

minus15

minus20

(b)

minus40 minus20 20 40

2

minus8

minus10

minus6

minus4

minus2

(c)

15

1

05

minus30 minus20 minus10 10 20 30

minus05

minus1

(d)

Figure 6 The varying process for graphs of 11990614(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

Journal of Applied Mathematics 7

minus10minus15 minus5 5 10 15minus01

minus02

minus03

minus04

minus05

minus06

minus07

(a) 119906 = 11990611(120585)

14

12

1

08

06

04

02

minus40 minus20 20 40

(b) 119906 = 11990612(120585)

60

50

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990615(120585)

50

40

30

20

10

minus20 minus10 10 20

(d) 119906 = 11990616(120585)

Figure 7 The graphs of 11990611(120585) 11990612(120585) when 119888 = 2 119892 = 4radic527 and 119906

15(120585) 11990616(120585) when 119888 = minus1 119892 = minus10

and its traveling wave equation

(120572 minus 119888) 120595 + 1205731205954+ 2120574119888(120595

1015840)2

+ 212057411988812059512059510158401015840= 119892 (33)

For given constants 119888 and 119892 there are the followingresults

(1∘) When 119888 gt 120572 and 119892 = 0 (32) has two elliptic periodicwave solutions

11990617(120585) =

1 minus cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 + (radic3 minus 1) cn (120578

5120585 1198965))

11990618(120585) =

1 + cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 minus (radic3 minus 1) cn (120578

5120585 1198965))

(34)

where

1205785=4radic3radic

119888 minus 120572

3120574119888

6radic

120573

2 (119888 minus 120572)

1198965=radic3 minus 1

2radic2

(35)

(2∘) When 119888 lt 0 and 119892 = minus 3radic(119888 minus 120572)416120573 (32) has twohyperbolic blow-up solutions

11990619(120585) =

3radic119888 minus 120572 (minus4 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (2 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

11990620(120585) =

3radic119888 minus 120572 (4 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (minus2 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

(36)

where

1205786= minusradic

120573

minus120574119888

3radic119888 minus 120572

2120573 (37)

For the graphs of 119906119894(120585) (119894 = 17ndash20) see Figure 8

Remark 7 In order to confirm the correctness of thesesolutions we have verified them by using the softwareMathematica for instance about 119906

20(120585) the commands are as

follows

120585 = 119909 minus 119888119905

120578 = minusradic120573

minus120574119888

3radic119888 minus 120572

3radic2120573

8 Journal of Applied Mathematics

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(a) 119906 = 11990617(120585)

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(b) 119906 = 11990618(120585)

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990619(120585)

minus40

minus30

minus20

minus10

minus20 minus10 10 20

(d) 119906 = 11990620(120585)

Figure 8 The graphs of 11990617(120585) 11990618(120585) when 119888 = 20 119892 = 0 and 119906

19(120585) 11990620(120585) when 119888 = minus16

119906 =

3radic119888 minus 120572 (4 + 2 cosh [120578120585] + radic6 sinh [120578120585])3radic2120573 (minus2 + 2 cosh [120578120585] + radic6 sinh [120578120585])

Simplify [119863 [119906 119905] + 120572119863 [119906 119909]

+ 120573119863 [1199064 119909] minus 120574119863 [119906

2 119909 2 119905]]

0

(38)

3 The Derivations for Proposition 1

In this section firstly we derive the precise expressions ofthe traveling wave solutions for BBM-like 119861(3 2) equationSecondly we show the relationships among these solutionstheoretically Substituting 119906(119909 119905) = 120593(120585) with 120585 = 119909 minus 119888119905 into(9) it follows that

(120572 minus 119888) 1205931015840+ 120573(120593

3)1015840

+ 120574119888(1205932)101584010158401015840

= 0 (39)

Integrating (39) once we have

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (40)

where 119892 is an integral constant

Letting 119910 = 1205931015840 we obtain the following planar system

119889120593

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

2120574119888120593

(41)

Under the transformation

119889120585 = 2120574119888120593119889120591 (42)

system (41) becomes

119889120593

119889120591= 2120574119888120593119910

119889119910

119889120591= 119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

(43)

Clearly system (41) and system (43) have the same firstintegral

12057411988812059321199102minus119892

21205932minus119888 minus 120572

31205933+120573

51205935= ℎ (44)

where ℎ is an integral constant Consequently these twosystems have the same topological phase portraits except for

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(a)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(b)

minus40 minus20 20 40

minus20

minus40

minus60

minus80

(c)

minus10 10 20 30

minus20

minus40

minus20minus30

minus60

minus80

(d)

Figure 3 The varying process for graphs of 1199065(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

1

05

minus20 minus10 10 20

(a)

1

15

05

minus05

minus20 minus10 10 20

(b)

1

15

05

minus05

minus20 minus10 10 20

(c)

1

15

05

minus05

minus20 minus10 10 20

(d)

Figure 4 The varying process for graphs of 1199067(120585) when 119888 gt 120572 and 119892 rarr 119892

0minus 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0minus 10minus1 (b)

119892 = 1198920minus 10minus2 (c) 119892 = 119892

0minus 10minus4 (d) 119892 = 119892

0minus 10minus6

6 Journal of Applied Mathematics

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(b)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(c)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(d)

Figure 5 The varying process for graphs of 11990613(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus5

minus10

minus15

minus20

(b)

minus40 minus20 20 40

2

minus8

minus10

minus6

minus4

minus2

(c)

15

1

05

minus30 minus20 minus10 10 20 30

minus05

minus1

(d)

Figure 6 The varying process for graphs of 11990614(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

Journal of Applied Mathematics 7

minus10minus15 minus5 5 10 15minus01

minus02

minus03

minus04

minus05

minus06

minus07

(a) 119906 = 11990611(120585)

14

12

1

08

06

04

02

minus40 minus20 20 40

(b) 119906 = 11990612(120585)

60

50

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990615(120585)

50

40

30

20

10

minus20 minus10 10 20

(d) 119906 = 11990616(120585)

Figure 7 The graphs of 11990611(120585) 11990612(120585) when 119888 = 2 119892 = 4radic527 and 119906

15(120585) 11990616(120585) when 119888 = minus1 119892 = minus10

and its traveling wave equation

(120572 minus 119888) 120595 + 1205731205954+ 2120574119888(120595

1015840)2

+ 212057411988812059512059510158401015840= 119892 (33)

For given constants 119888 and 119892 there are the followingresults

(1∘) When 119888 gt 120572 and 119892 = 0 (32) has two elliptic periodicwave solutions

11990617(120585) =

1 minus cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 + (radic3 minus 1) cn (120578

5120585 1198965))

11990618(120585) =

1 + cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 minus (radic3 minus 1) cn (120578

5120585 1198965))

(34)

where

1205785=4radic3radic

119888 minus 120572

3120574119888

6radic

120573

2 (119888 minus 120572)

1198965=radic3 minus 1

2radic2

(35)

(2∘) When 119888 lt 0 and 119892 = minus 3radic(119888 minus 120572)416120573 (32) has twohyperbolic blow-up solutions

11990619(120585) =

3radic119888 minus 120572 (minus4 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (2 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

11990620(120585) =

3radic119888 minus 120572 (4 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (minus2 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

(36)

where

1205786= minusradic

120573

minus120574119888

3radic119888 minus 120572

2120573 (37)

For the graphs of 119906119894(120585) (119894 = 17ndash20) see Figure 8

Remark 7 In order to confirm the correctness of thesesolutions we have verified them by using the softwareMathematica for instance about 119906

20(120585) the commands are as

follows

120585 = 119909 minus 119888119905

120578 = minusradic120573

minus120574119888

3radic119888 minus 120572

3radic2120573

8 Journal of Applied Mathematics

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(a) 119906 = 11990617(120585)

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(b) 119906 = 11990618(120585)

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990619(120585)

minus40

minus30

minus20

minus10

minus20 minus10 10 20

(d) 119906 = 11990620(120585)

Figure 8 The graphs of 11990617(120585) 11990618(120585) when 119888 = 20 119892 = 0 and 119906

19(120585) 11990620(120585) when 119888 = minus16

119906 =

3radic119888 minus 120572 (4 + 2 cosh [120578120585] + radic6 sinh [120578120585])3radic2120573 (minus2 + 2 cosh [120578120585] + radic6 sinh [120578120585])

Simplify [119863 [119906 119905] + 120572119863 [119906 119909]

+ 120573119863 [1199064 119909] minus 120574119863 [119906

2 119909 2 119905]]

0

(38)

3 The Derivations for Proposition 1

In this section firstly we derive the precise expressions ofthe traveling wave solutions for BBM-like 119861(3 2) equationSecondly we show the relationships among these solutionstheoretically Substituting 119906(119909 119905) = 120593(120585) with 120585 = 119909 minus 119888119905 into(9) it follows that

(120572 minus 119888) 1205931015840+ 120573(120593

3)1015840

+ 120574119888(1205932)101584010158401015840

= 0 (39)

Integrating (39) once we have

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (40)

where 119892 is an integral constant

Letting 119910 = 1205931015840 we obtain the following planar system

119889120593

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

2120574119888120593

(41)

Under the transformation

119889120585 = 2120574119888120593119889120591 (42)

system (41) becomes

119889120593

119889120591= 2120574119888120593119910

119889119910

119889120591= 119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

(43)

Clearly system (41) and system (43) have the same firstintegral

12057411988812059321199102minus119892

21205932minus119888 minus 120572

31205933+120573

51205935= ℎ (44)

where ℎ is an integral constant Consequently these twosystems have the same topological phase portraits except for

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(b)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(c)

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(d)

Figure 5 The varying process for graphs of 11990613(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

minus40 minus20 20 40

minus10

minus20

minus30

minus40

(a)

minus40 minus20 20 40

minus5

minus10

minus15

minus20

(b)

minus40 minus20 20 40

2

minus8

minus10

minus6

minus4

minus2

(c)

15

1

05

minus30 minus20 minus10 10 20 30

minus05

minus1

(d)

Figure 6 The varying process for graphs of 11990614(120585) when 119888 gt 120572 and 119892 rarr 119892

0+ 0 where 120572 = 120573 = 120574 = 1 119888 = 2 and (a) 119892 = 119892

0+ 15 (b)

119892 = 1198920+ 10minus2 (c) 119892 = 119892

0+ 10minus6 (d) 119892 = 119892

0+ 10minus8

Journal of Applied Mathematics 7

minus10minus15 minus5 5 10 15minus01

minus02

minus03

minus04

minus05

minus06

minus07

(a) 119906 = 11990611(120585)

14

12

1

08

06

04

02

minus40 minus20 20 40

(b) 119906 = 11990612(120585)

60

50

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990615(120585)

50

40

30

20

10

minus20 minus10 10 20

(d) 119906 = 11990616(120585)

Figure 7 The graphs of 11990611(120585) 11990612(120585) when 119888 = 2 119892 = 4radic527 and 119906

15(120585) 11990616(120585) when 119888 = minus1 119892 = minus10

and its traveling wave equation

(120572 minus 119888) 120595 + 1205731205954+ 2120574119888(120595

1015840)2

+ 212057411988812059512059510158401015840= 119892 (33)

For given constants 119888 and 119892 there are the followingresults

(1∘) When 119888 gt 120572 and 119892 = 0 (32) has two elliptic periodicwave solutions

11990617(120585) =

1 minus cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 + (radic3 minus 1) cn (120578

5120585 1198965))

11990618(120585) =

1 + cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 minus (radic3 minus 1) cn (120578

5120585 1198965))

(34)

where

1205785=4radic3radic

119888 minus 120572

3120574119888

6radic

120573

2 (119888 minus 120572)

1198965=radic3 minus 1

2radic2

(35)

(2∘) When 119888 lt 0 and 119892 = minus 3radic(119888 minus 120572)416120573 (32) has twohyperbolic blow-up solutions

11990619(120585) =

3radic119888 minus 120572 (minus4 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (2 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

11990620(120585) =

3radic119888 minus 120572 (4 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (minus2 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

(36)

where

1205786= minusradic

120573

minus120574119888

3radic119888 minus 120572

2120573 (37)

For the graphs of 119906119894(120585) (119894 = 17ndash20) see Figure 8

Remark 7 In order to confirm the correctness of thesesolutions we have verified them by using the softwareMathematica for instance about 119906

20(120585) the commands are as

follows

120585 = 119909 minus 119888119905

120578 = minusradic120573

minus120574119888

3radic119888 minus 120572

3radic2120573

8 Journal of Applied Mathematics

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(a) 119906 = 11990617(120585)

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(b) 119906 = 11990618(120585)

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990619(120585)

minus40

minus30

minus20

minus10

minus20 minus10 10 20

(d) 119906 = 11990620(120585)

Figure 8 The graphs of 11990617(120585) 11990618(120585) when 119888 = 20 119892 = 0 and 119906

19(120585) 11990620(120585) when 119888 = minus16

119906 =

3radic119888 minus 120572 (4 + 2 cosh [120578120585] + radic6 sinh [120578120585])3radic2120573 (minus2 + 2 cosh [120578120585] + radic6 sinh [120578120585])

Simplify [119863 [119906 119905] + 120572119863 [119906 119909]

+ 120573119863 [1199064 119909] minus 120574119863 [119906

2 119909 2 119905]]

0

(38)

3 The Derivations for Proposition 1

In this section firstly we derive the precise expressions ofthe traveling wave solutions for BBM-like 119861(3 2) equationSecondly we show the relationships among these solutionstheoretically Substituting 119906(119909 119905) = 120593(120585) with 120585 = 119909 minus 119888119905 into(9) it follows that

(120572 minus 119888) 1205931015840+ 120573(120593

3)1015840

+ 120574119888(1205932)101584010158401015840

= 0 (39)

Integrating (39) once we have

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (40)

where 119892 is an integral constant

Letting 119910 = 1205931015840 we obtain the following planar system

119889120593

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

2120574119888120593

(41)

Under the transformation

119889120585 = 2120574119888120593119889120591 (42)

system (41) becomes

119889120593

119889120591= 2120574119888120593119910

119889119910

119889120591= 119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

(43)

Clearly system (41) and system (43) have the same firstintegral

12057411988812059321199102minus119892

21205932minus119888 minus 120572

31205933+120573

51205935= ℎ (44)

where ℎ is an integral constant Consequently these twosystems have the same topological phase portraits except for

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

minus10minus15 minus5 5 10 15minus01

minus02

minus03

minus04

minus05

minus06

minus07

(a) 119906 = 11990611(120585)

14

12

1

08

06

04

02

minus40 minus20 20 40

(b) 119906 = 11990612(120585)

60

50

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990615(120585)

50

40

30

20

10

minus20 minus10 10 20

(d) 119906 = 11990616(120585)

Figure 7 The graphs of 11990611(120585) 11990612(120585) when 119888 = 2 119892 = 4radic527 and 119906

15(120585) 11990616(120585) when 119888 = minus1 119892 = minus10

and its traveling wave equation

(120572 minus 119888) 120595 + 1205731205954+ 2120574119888(120595

1015840)2

+ 212057411988812059512059510158401015840= 119892 (33)

For given constants 119888 and 119892 there are the followingresults

(1∘) When 119888 gt 120572 and 119892 = 0 (32) has two elliptic periodicwave solutions

11990617(120585) =

1 minus cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 + (radic3 minus 1) cn (120578

5120585 1198965))

11990618(120585) =

1 + cn (1205785120585 1198965)

3radic1205732 (119888 minus 120572) (1 + radic3 minus (radic3 minus 1) cn (120578

5120585 1198965))

(34)

where

1205785=4radic3radic

119888 minus 120572

3120574119888

6radic

120573

2 (119888 minus 120572)

1198965=radic3 minus 1

2radic2

(35)

(2∘) When 119888 lt 0 and 119892 = minus 3radic(119888 minus 120572)416120573 (32) has twohyperbolic blow-up solutions

11990619(120585) =

3radic119888 minus 120572 (minus4 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (2 minus 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

11990620(120585) =

3radic119888 minus 120572 (4 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

3radic2120573 (minus2 + 2 cosh (120578

6120585) + radic6 sinh (120578

6

10038161003816100381610038161205851003816100381610038161003816))

(36)

where

1205786= minusradic

120573

minus120574119888

3radic119888 minus 120572

2120573 (37)

For the graphs of 119906119894(120585) (119894 = 17ndash20) see Figure 8

Remark 7 In order to confirm the correctness of thesesolutions we have verified them by using the softwareMathematica for instance about 119906

20(120585) the commands are as

follows

120585 = 119909 minus 119888119905

120578 = minusradic120573

minus120574119888

3radic119888 minus 120572

3radic2120573

8 Journal of Applied Mathematics

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(a) 119906 = 11990617(120585)

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(b) 119906 = 11990618(120585)

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990619(120585)

minus40

minus30

minus20

minus10

minus20 minus10 10 20

(d) 119906 = 11990620(120585)

Figure 8 The graphs of 11990617(120585) 11990618(120585) when 119888 = 20 119892 = 0 and 119906

19(120585) 11990620(120585) when 119888 = minus16

119906 =

3radic119888 minus 120572 (4 + 2 cosh [120578120585] + radic6 sinh [120578120585])3radic2120573 (minus2 + 2 cosh [120578120585] + radic6 sinh [120578120585])

Simplify [119863 [119906 119905] + 120572119863 [119906 119909]

+ 120573119863 [1199064 119909] minus 120574119863 [119906

2 119909 2 119905]]

0

(38)

3 The Derivations for Proposition 1

In this section firstly we derive the precise expressions ofthe traveling wave solutions for BBM-like 119861(3 2) equationSecondly we show the relationships among these solutionstheoretically Substituting 119906(119909 119905) = 120593(120585) with 120585 = 119909 minus 119888119905 into(9) it follows that

(120572 minus 119888) 1205931015840+ 120573(120593

3)1015840

+ 120574119888(1205932)101584010158401015840

= 0 (39)

Integrating (39) once we have

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (40)

where 119892 is an integral constant

Letting 119910 = 1205931015840 we obtain the following planar system

119889120593

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

2120574119888120593

(41)

Under the transformation

119889120585 = 2120574119888120593119889120591 (42)

system (41) becomes

119889120593

119889120591= 2120574119888120593119910

119889119910

119889120591= 119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

(43)

Clearly system (41) and system (43) have the same firstintegral

12057411988812059321199102minus119892

21205932minus119888 minus 120572

31205933+120573

51205935= ℎ (44)

where ℎ is an integral constant Consequently these twosystems have the same topological phase portraits except for

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(a) 119906 = 11990617(120585)

3

25

2

15

1

05

minus30 minus20 minus10 10 20 30

(b) 119906 = 11990618(120585)

40

30

20

10

minus20 minus10 10 20

(c) 119906 = 11990619(120585)

minus40

minus30

minus20

minus10

minus20 minus10 10 20

(d) 119906 = 11990620(120585)

Figure 8 The graphs of 11990617(120585) 11990618(120585) when 119888 = 20 119892 = 0 and 119906

19(120585) 11990620(120585) when 119888 = minus16

119906 =

3radic119888 minus 120572 (4 + 2 cosh [120578120585] + radic6 sinh [120578120585])3radic2120573 (minus2 + 2 cosh [120578120585] + radic6 sinh [120578120585])

Simplify [119863 [119906 119905] + 120572119863 [119906 119909]

+ 120573119863 [1199064 119909] minus 120574119863 [119906

2 119909 2 119905]]

0

(38)

3 The Derivations for Proposition 1

In this section firstly we derive the precise expressions ofthe traveling wave solutions for BBM-like 119861(3 2) equationSecondly we show the relationships among these solutionstheoretically Substituting 119906(119909 119905) = 120593(120585) with 120585 = 119909 minus 119888119905 into(9) it follows that

(120572 minus 119888) 1205931015840+ 120573(120593

3)1015840

+ 120574119888(1205932)101584010158401015840

= 0 (39)

Integrating (39) once we have

(120572 minus 119888) 120593 + 1205731205933+ 2120574119888(120593

1015840)2

+ 212057411988812059312059310158401015840= 119892 (40)

where 119892 is an integral constant

Letting 119910 = 1205931015840 we obtain the following planar system

119889120593

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

2120574119888120593

(41)

Under the transformation

119889120585 = 2120574119888120593119889120591 (42)

system (41) becomes

119889120593

119889120591= 2120574119888120593119910

119889119910

119889120591= 119892 + (119888 minus 120572) 120593 minus 120573120593

3minus 2120574119888119910

2

(43)

Clearly system (41) and system (43) have the same firstintegral

12057411988812059321199102minus119892

21205932minus119888 minus 120572

31205933+120573

51205935= ℎ (44)

where ℎ is an integral constant Consequently these twosystems have the same topological phase portraits except for

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

yΓ1

1205931 o 120593120593lowast1

(a) 0 lt 119888 lt 120572 119892 lt 0

yΓ1

o 120593120593lowast1 = 1205931 = 0

(b) 0 lt 119888 lt 120572 119892 = 0

y

Γ1

1205931o 120593120593lowast1

(c) 0 lt 119888 lt 120572 119892 gt 0

yΓ1

1205931 o 120593120593lowast1 120593lowast2 120593lowast3

(d) 119888 gt 120572 119892 le minus1198920

yΓ1

Γ2

1205931 1205932 1205933o 120593120593lowast1 120593lowast2 120593lowast3

(e) 119888 gt 120572 minus1198920lt 119892 lt 0

yΓ1

Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 = 120593lowast2 = 0

(f) 119888 gt 120572 119892 = 0

yΓ1 Γ2

1205931 1205933o 120593120593lowast1 120593lowast31205932 120593lowast2

(g) 119888 gt 120572 0 lt 119892 lt 1198920

yΓ3 Γ4

1205933 120593120593lowast2 120593lowast3120593lowast1 = 1205931 = 1205932 o

(h) 119888 gt 120572 119892 = 1198920

yΓ5

1205933o 120593120593lowast1 120593lowast3120593lowast2

(i) 119888 gt 120572 119892 gt 1198920

Figure 9 The phase portraits of system (43) when 119888 gt 0 and 119888 = 120572

y

Γ6

1205931 o 120593120593lowast1

(a) 119892 lt 0

y

Γ6

o 120593120593lowast1 = 1205931 = 0

(b) 119892 = 0

yΓ6

1205931o 120593120593lowast1

(c) 119892 gt 0

Figure 10 The phase portraits of system (43) when 119888 lt 0

the straight line 120593 = 0 Thus we can understand the phaseportraits of system (41) from that of system (43)

When the integral constant ℎ = 0 (44) becomes

12057411988812059321199102minus 1205932(119892

2+119888 minus 120572

3120593 minus

120573

51205933) = 0 (45)

Solving equation 1198922+ ((119888 minus120572)3)120593minus (1205735)1205933 = 0 we getthree roots 120593

1 1205932 and 120593

3as (15) (16) and (17)

On the other hand solving equation

119910 = 0

119892 + (119888 minus 120572) 120593 minus 1205731205933minus 2120574119888119910

2= 0

(46)

we get three three singular points (120593lowast119894 0) (119894 = 1 2 3) where

120593lowast

1=21205733radic18 (120572 minus 119888) minus

3radic12Δ

23

6120573Δ13

120593lowast

2=

21205733radic26radic3 (radic3 minus 3119894) (119888 minus 120572) +

3radic46radic9 (1 + radic3119894) Δ

23

12120573Δ13

120593lowast

3=

21205733radic26radic3 (radic3 + 3119894) (119888 minus 120572) +

3radic46radic9 (1 minus radic3119894) Δ

23

12120573Δ13

Δ = radic8111989221205734 minus 12(119888 minus 120572)31205733 minus 9119892120573

2

(47)According to the qualitative theory we obtain the phase

portraits of system (43) as Figures 9 and 10

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

From Figures 9 and 10 one can see that there are six kindsof orbits Γ

119894(119894 = 1ndash6) where Γ

1(Γ6) passes (120593

1 0) Γ2passes

(1205932 0) and (120593

3 0) Γ3and Γ4pass (120593

1 0) and (120593

3 0) and Γ

5

passes (1205933 0) Now we will derive the explicit expressions of

solutions for the BBM-like 119861(3 2) equation respectively(1) When 0 lt 119888 lt 120572 or 119888 gt 120572 119892 lt minus119892

0 Γ1has the

expression

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593 minus 120593

2) (120593 minus 120593

3) 120593 le 120593

1

(48)

where 1205932and 120593

3are complex numbers

Substituting (48) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(49)

Completing the integrals in the above two equations andnoting that 119906 = 120593(120585) we obtain 119906

1(120585) and 119906

2(120585) as (11)

(2)When 119888 gt 120572 and 119892 = minus1198920 Γ1has the expression

Γ1 119910 = plusmnradic

120573

5120574119888(1205932minus 120593)radic120593

1minus 120593 120593 le 120593

1 (50)

where 1205931= minus2radic5(119888 minus 120572)9120573 120593

2= radic5(119888 minus 120572)9120573

Substituting (50) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

(1205932minus 119904)radic1205931 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(51)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

3(120585) and 119906

4(120585) as (19) and

(20)(3) When 119888 gt 120572 and minus119892

0lt 119892 lt 119892

0 Γ1and Γ2have the

expressions

Γ1 119910 = plusmnradic

120573

5120574119888radic(1205931minus 120593) (120593

2minus 120593) (120593

3minus 120593) 120593 le 120593

1

Γ2 119910 = plusmnradic

120573

5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (1205933minus 120593) 120593

2le 120593 le 120593

3

(52)

Substituting (52) into 119889120593119889120585 = 119910 and integrating themwe have

int

120593

minusinfin

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

120593

119889119904

radic(1205931minus 119904) (120593

2minus 119904) (120593

3minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205932

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (1205933minus 119904)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(53)

Completing the integrals in the above four equations andnoting that 119906 = 120593(120585) we obtain 119906

119894(120585) (119894 = 5ndash8) as (21)

(4)When 119888 gt 120572 and119892 = 1198920 Γ3and Γ4have the expressions

Γ3 119910 = plusmnradic

120573

5120574119888(1205931minus 120593)radic120593

3minus 120593 120593 lt 120593

1

Γ4 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 le 120593

3

(54)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (54) into 119889120593119889120585 = 119910 and integrating themwe have

int

1205933

120593

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205931

minusinfin

119889119904

(1205931minus 119904)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(55)

Completing the integrals in the two equations above andnoting that 119906 = 120593(120585) we obtain 119906

9(120585) and 119906

10(120585) as (23) and

(24)(5)When 119888 gt 120572 and 119892

0lt 119892 Γ

5has the expression

Γ5 119910 = plusmnradic

120573

5120574119888radic(1205933minus 120593) (120593 minus 120593

2) (120593 minus 120593

1) 120593 le 120593

3

(56)

where 1205931and 120593

2are complex numbers

Substituting (56) into 119889120593119889120585 = 119910 and integrating it wehave

int

120593

minusinfin

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

1205933

120593

119889119904

radic(1205933minus 119904) (119904 minus 120593

2) (119904 minus 120593

1)

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(57)

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

(a)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(b)

minus30 minus20 minus10 10 20 30minus01

minus02

minus03

minus04

minus05

minus06

minus07

(c)

Γ7

1205931 o 120593120593lowast2

(d)

Figure 11 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (a) initial value 120593(0) = minus02 (b) initial value120593(0) = minus001 and (c) initial value 120593(0) = minus000001

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

13(120585) and 119906

14(120585) as (28)

(6)When 119888 lt 0 Γ6has the expression

Γ6 119910 = plusmnradic

120573

minus5120574119888radic(120593 minus 120593

1) (120593 minus 120593

2) (120593 minus 120593

3) 120593

1le 120593

(58)

where 1205932and 120593

3are complex numbers

Substituting (58) into 119889120593119889120585 = 119910 and integrating it wehave

int

+infin

120593

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

int

120593

1205931

119889119904

radic(119904 minus 1205931) (119904 minus 120593

2) (119904 minus 120593

3)

= radic120573

minus5120574119888

10038161003816100381610038161205851003816100381610038161003816

(59)

Completing the integrals in the two above equations andnoting that 119906 = 120593(120585) we obtain 119906

15(120585) and 119906

16(120585) as (30)

(7) When 119888 gt 120572 and 119892 = 1198920 there are two special

kinds of orbits Γ7surrounding the center point (120593lowast

2 0) (see

Figure 11(d)) and Γ8surrounding the center point (120593lowast

3 0) (see

Figure 12(a)) which are the boundaries of two families of

closed orbits Note that the periodic waves of (9) correspondto the periodic integral curves of (40) and the periodicintegral curves correspond to the closed orbits of system (41)For given constants 119888 119892 and the corresponding initial value120593(0) we simulate the integral curves of (40) as shown inFigures 11 and 12

From Figure 11 we see that when the initial value 120593(0)tends to 0 minus 0 the periodic integral curve tends to peakonThis implies that the orbit Γ

7corresponds to peakon On 120593minus119910

plane Γ7has the expression

Γ7 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 120593

1lt 120593 lt 0 (60)

where 1205931= minusradic5(119888 minus 120572)9120573 120593

3= 2radic5(119888 minus 120572)9120573

Substituting (60) into 119889120593119889120585 = 119910 and integrating it wehave

int

0

120593

119889119904

(119904 minus 1205931)radic1205932 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(61)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

11(120585) as (25)

Similarly from Figure 12 we see that when the initialvalue 120593(0) tends to 0 + 0 the periodic integral curve tends to

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

Γ8

1205933o 120593120593lowast3

(a)

minus20 minus10 100 20

14

12

1

08

06

04

02

(b)

minus20 minus10 100 20

14

12

1

08

06

04

02

(c)

minus20 minus10 100 20

14

12

1

08

06

04

02

(d)

Figure 12 The simulation of integral curves of (40) when 120572 = 120573 = 120574 = 1 119888 = 2 119892 = 4radic527 (b) initial value 120593(0) = 05 (c) initial value120593(0) = 01 and (d) initial value 120593(0) = 00001

a periodic peakon This implies that the orbit Γ8corresponds

to a periodic peakon On 120593 minus 119910 plane Γ8has the expression

Γ8 119910 = plusmnradic

120573

5120574119888(120593 minus 120593

1)radic1205933minus 120593 0 lt 120593 le 120593

3 (62)

Substituting (62) into 119889120593119889120585 = 119910 and integrating it we have

int

120593

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

= radic120573

5120574119888

10038161003816100381610038161205851003816100381610038161003816

(63)

Completing the integral in the above equation andnoting that119906 = 120593(120585) we obtain 119906

12(120585) as (26) where

119879 = radic5120574119888

120573

100381610038161003816100381610038161003816100381610038161003816

int

1205933

0

119889119904

(119904 minus 1205931)radic1205933 minus 119904

100381610038161003816100381610038161003816100381610038161003816

= radic5120574119888

120573

4radic

120573

5 (119888 minus 120572)

10038161003816100381610038161003816ln (5 minus 2radic6)10038161003816100381610038161003816

(64)

Hereto we have finished the derivations for the solutions119906119894(120585) (119894 = 1ndash16) In what follows we shall derive the

relationships among these solutions

(1∘) When 119888 gt 120572 and 119892 rarr minus1198920 it follows that

1205931997888rarr minus2radic

5 (119888 minus 120572)

9120573

1205932997888rarr radic

5 (119888 minus 120572)

9120573

1205933997888rarr radic

5 (119888 minus 120572)

9120573

1198601997888rarr radic

5 (119888 minus 120572)

120573

1205781997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198961997888rarr 0

cn (1205781120585 1198961) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 0)

= cos(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 13

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 0

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= sin(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

(65)Thus we have

1199061(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573minus

2radic5 (119888 minus 120572) 120573

1 minus cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199062(120585) 997888rarr minus2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 0)

= radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + cos (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sec2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199065(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573snminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573sinminus2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573csc2(radic

120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199063(120585) (see (19))

1199066(120585) 997888rarr ( minus 2radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0))

minus1

= ( minus 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573

times sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times(1 minus sin2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= ( minus radic5 (119888 minus 120572)

120573+ radic

5 (119888 minus 120572)

9120573

times cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (cos2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times sec2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199064(120585) (see (20))

1199067(120585) 997888rarr radic

5 (119888 minus 120572)

9120573minus (radic

5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 0)

= radic5 (119888 minus 120572)

9120573

1199068(120585) 997888rarr

radic5 (119888 minus 120572) 9120573 minus 0

1 minus 0

= radic5 (119888 minus 120572)

9120573

(66)

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Journal of Applied Mathematics

(2∘) When 119888 gt 120572 and 119892 rarr 1198920 it follows that

1205931997888rarr minusradic

5 (119888 minus 120572)

9120573

1205932997888rarr minusradic

5 (119888 minus 120572)

9120573

1205933997888rarr 2radic

5 (119888 minus 120572)

9120573

1205782997888rarr radic

120573

120574119888

4radic119888 minus 120572

80120573

1198962997888rarr 1

sn (1205782120585 1198962) 997888rarr sn(radic

120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= tanh(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

1198602997888rarr radic

5 (119888 minus 120572)

120573

1205783997888rarr radic

120573

120574119888

4radic119888 minus 120572

5120573

1198963997888rarr 1

cn (1205783120585 1198963) 997888rarr cn(radic

120573

120574119888

4radic119888 minus 120572

5120573120585 1)

= sech(radic120573

120574119888

4radic119888 minus 120572

5120573120585)

(67)

Thus we have

1199065(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times snminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanhminus2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 + csch2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 11990610(120585) (see (24))

1199066(120585) (or 119906

8(120585)) 997888rarr (minus radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

times (1 minus sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1))

minus1

= minusradic5 (119888 minus 120572)

9120573

1199067(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

9120573)

times sn2(radic120573

120574119888

4radic119888 minus 120572

80120573120585 1)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times tanh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 2radic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times (1 minus sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

= 1199069(120585) (see (23))

11990613(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 minus cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 15

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 minus sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times(cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) minus 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2sinh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573minus radic

5 (119888 minus 120572)

120573

times csch2(radic120573

120574119888

4radic(119888 minus 120572)

80120573120585)

= 11990610(120585) (see (24))

11990614(120585) 997888rarr 2radic

5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

minus2radic5 (119888 minus 120572) 120573

1 + cn (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585 1)

= 5radic5 (119888 minus 120572)

9120573

minus2radic5 (119888 minus 120572) 120573

1 + sech (radic120573120574119888 4radic(119888 minus 120572) 5120573 120585)

= 5radic5 (119888 minus 120572)

9120573

minus (2radic5 (119888 minus 120572)

120573

times cosh (radic120573

120574119888

4radic119888 minus 120572

5120573120585))

times (cosh(radic120573

120574119888

4radic119888 minus 120572

5120573120585) + 1)

minus1

= 5radic5 (119888 minus 120572)

9120573

minus (minus2radic5 (119888 minus 120572)

120573+ 4radic

5 (119888 minus 120572)

120573

times cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

times (2cosh2(radic120573

120574119888

4radic119888 minus 120572

80120573120585))

minus1

= minusradic5 (119888 minus 120572)

9120573+ radic

5 (119888 minus 120572)

120573

times sech2(radic120573

120574119888

4radic119888 minus 120572

80120573120585)

= 1199069(120585) (see (23))

(68)

Hereto we have completed the derivations forProposition 1

4 The Derivations for Proposition 6

In this section we derive the precise expressions of the trav-eling wave solutions for BBM-like119861(4 2) equation Similar tothe derivations in Section 3 substituting 119906(119909 119905) = 120595(120585) with120585 = 119909 minus 119888119905 into (32) and integrating it we have the followingplanar system

119889120595

119889120585= 119910

119889119910

119889120585=119892 + (119888 minus 120572) 120595 minus 120573120595

4minus 2120574119888119910

2

2120574119888120595

(69)

with the first integral

12057411988812059521199102minus119892

21205952minus119888 minus 120572

31205953+120573

61205956= ℎ (70)

where 119892 and ℎ are the integral constants

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Journal of Applied Mathematics

When 119888 gt 120572 ℎ = 0 and 119892 = 0 (70) becomes

119910 = plusmnradic119888 minus 120572

3120574119888radic120595(1 minus

120573

2 (119888 minus 120572)1205953) 0 le 120595 le

3radic2 (119888 minus 120572)

120573

(71)

Substituting (71) into the first equation of (69) andintegrating it from 0 to 120595 or 120595 to 3radic2(119888 minus 120572)120573 respectivelyit follows that

int

120595

0

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

int

3radic2(119888minus120572)120573

120595

119889119904

radic119904 (1 minus (1205732 (119888 minus 120572)) 1199043)

= radic119888 minus 120572

3120574119888

10038161003816100381610038161205851003816100381610038161003816

(72)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two elliptic periodic wavesolutions 119906

17(120585) and 119906

18(120585) as (34)

When 119888 lt 0 ℎ = 0 and 119892 = minus3radic(119888 minus 120572)

416120573 (70)

becomes

119910 = plusmnradic120573

minus6120574119888

1003816100381610038161003816120595 minus 1199011003816100381610038161003816radic1205952 + 119902120595 + 119903 (73)

where

119901 = 3radic119888 minus 120572

2120573 119902 =

3radic4 (119888 minus 120572)

120573 119903 =

3radic27(119888 minus 120572)

2

41205732

(74)

Substituting (73) into the first equation of (69) andintegrating it from minusinfin to 120595 or 120595 to +infin respectively itfollows that

int

120595

minusinfin

119889119904

(119901 minus 119904)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (120595 lt 119901)

int

+infin

120595

119889119904

(119904 minus 119901)radic1199042 + 119902119904 + 119903

=120573

radicminus6120574119888

10038161003816100381610038161205851003816100381610038161003816 (119901 lt 120595)

(75)

Completing the integrals in the two above equations andnoting that 119906 = 120595(120585) we obtain the two hyperbolic blow-upsolutions 119906

19(120585) and 119906

20(120585) as (36)

Hereto we have completed the derivations forProposition 6

5 Conclusion

In this paper we have investigated the nonlinear wavesolutions and their bifurcations for BBM-like 119861(119898 2) (119898 =

3 4) equations For BBM-like 119861 (3 2) equation we obtainsome precise expressions of traveling wave solutions (see119906119894(120585) (119894 = 1ndash16)) which include periodic blow-up and

periodic wave solution peakon and periodic peakon wavesolution and solitary wave and blow-up solution We also

reveal the relationships among these solutions theoretically(see Remarks 2ndash5 and the corresponding derivations) ForBBM-like 119861(4 2) equation we construct two elliptic periodicwave solutions and two hyperbolic blow-up solutions (see119906119894(120585) (119894 = 17ndash20)) We would like to study the BBM-like

119861(119898 119899) equations further

Conflict of Interests

The authors declare that they do not have any commercialor associative interest that represents a conflict of interests inconnection with the work submitted

Acknowledgment

This work is supported by the National Natural ScienceFoundation (no 11171115)

References

[1] G Liu and T Fan ldquoNew applications of developed Jacobi ellipticfunction expansionmethodsrdquo Physics Letters A vol 345 no 1-3pp 161ndash166 2005

[2] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[3] M A Abdou ldquoThe extended F-expansion method and itsapplication for a class of nonlinear evolution equationsrdquo ChaosSolitons and Fractals vol 31 no 1 pp 95ndash104 2007

[4] M Wang and X Li ldquoExtended F-expansion method and peri-odic wave solutions for the generalized Zakharov equationsrdquoPhysics Letters A vol 343 no 1-3 pp 48ndash54 2005

[5] M Wang X Li and J Zhang ldquoThe 1198661015840

119866-expansion methodand travelling wave solutions of nonlinear evolution equationsin mathematical physicsrdquo Physics Letters A vol 372 no 4 pp417ndash423 2008

[6] M Song and Y Ge ldquoApplication of the 1198661015840

119866-expansionmethodto (3+1)-dimensional nonlinear evolution equationsrdquoComput-ers and Mathematics with Applications vol 60 no 5 pp 1220ndash1227 2010

[7] J Li and Z Liu ldquoSmooth and non-smooth traveling waves ina nonlinearly dispersive equationrdquo Applied Mathematical Mod-elling vol 25 no 1 pp 41ndash56 2000

[8] Z Liu and Y Liang ldquoThe explicit nonlinear wave solutions andtheir bifurcations of the generalized camassaholm equationrdquoInternational Journal of Bifurcation and Chaos vol 21 no 11 pp3119ndash3136 2011

[9] Z Liu T Jiang P Qin and Q Xu ldquoTrigonometric functionperiodic wave solutions and their limit forms for the KdV-Like and the PC-Like equationsrdquo Mathematical Problems inEngineering vol 2011 Article ID 810217 23 pages 2011

[10] M Song S Li and J Cao ldquoNew exact solutions for the (2 +1) -dimensional Broer-Kaup-Kupershmidt equationsrdquo Abstractand Applied Analysis vol 2010 Article ID 652649 10 pages2010

[11] S Li and R Liu ldquoSome explicit expressions and interestingbifurcation phenomena for nonlinear waves in generalized zak-harov equationsrdquo Abstract and Applied Analysis vol 2013Article ID 869438 19 pages 2013

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 17

[12] D H Peregine ldquoCalculations of the development of an undularborerdquo Journal of Fluid Mechanics vol 25 pp 321ndash330 1964

[13] D H Peregine ldquoLong waves on a beachrdquo Journal of FluidMechanics vol 27 no 4 pp 815ndash827 1967

[14] T B Benjamin J L Bona and J Mahony ldquoModel equations forlongwave in nonlinear dispersive systems PhilosrdquoPhilosophicalTransactions of the Royal Society of London vol 272 no 1220pp 47ndash48 1972

[15] Y Shang ldquoExplicit and exact special solutions for BBM-like119861(119898 119899) equations with fully nonlinear dispersionrdquo ChaosSolitons and Fractals vol 25 no 5 pp 1083ndash1091 2005

[16] J Li andZ Liu ldquoTravelingwave solutions for a class of nonlineardispersive equationsrdquo Chinese Annals of Mathematics B vol 23no 3 pp 397ndash418 2002

[17] A M Wazwaz ldquoExact special solutions with solitary patternsfor the nonlinear dispersive119870(119898 119899) equationsrdquo Chaos Solitonsand Fractals vol 13 no 1 pp 161ndash170 2002

[18] R Liu ldquoSome new results on explicit traveling wave solutions of119870(119898 119899) equationrdquo Discrete and Continuous Dynamical SystemsB vol 13 no 3 pp 633ndash646 2010

[19] B Jiang Y Lu Q Ma Y Cao and Y Zhang ldquoBifurcation oftraveling wave solutions for the BBM-like B(2 2) equationrdquoApplied Mathematics and Computation vol 218 no 14 pp7375ndash7381 2012

[20] A-M Wazwaz ldquoAnalytic study on nonlinear variants of theRLW and the PHI-four equationsrdquo Communications in Nonlin-ear Science and Numerical Simulation vol 12 no 3 pp 314ndash3272007

[21] D Feng J Li J Lu and T He ldquoNew explicit and exact solutionsfor a system of variant RLW equationsrdquo Applied Mathematicsand Computation vol 198 no 2 pp 715ndash720 2008

[22] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

[23] S Kuru ldquoTraveling wave solutions of the BBM-like equationsrdquoJournal of Physics A vol 42 no 37 p 375203 12 2009

[24] S Kuru ldquoCompactons and kink-like solutions of BBM-likeequations by means of factorizationrdquo Chaos Solitons andFractals vol 42 no 1 pp 626ndash633 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of