Singularities Analysis and Traveling Wave Solutions for Generalized Hyperlastic-Rod Wave Equation

Guliang Cai, Zhenzhen Zhang Nonlinear Scientific Research Center

Faculty of Science, Jiangsu University Zhenjiang, P.R. China

e-mail: glcai@ujs.edu.cn, huashui.2008@163.com

Xianbin Wu Junior College

Zhejiang Wanli University Ningbo, Zhejiang, P.R. China e-mail: wxb3210@zwu.edu.cn

Abstract—In the paper, a kind of generalized hyperlastic-rod wave equation is studied. First the equation is transformed into the form of planar dynamic system by a series of transformations. Then the properties of equilibrium points and the orbits corresponding to them are studied by using the bifurcation theory of planar dynamic system. What’s more, topological phase portraits of the system are given. Through its first integral and combining with a new method, traveling wave solutions of the implicit form, index form and triangle function form of the equation are worked out.

Keywords- planar dynamic system; solutions of the index form; generalized hyperlastic-rod wave equation.

I. INTRODUCTION The research of the related properties of the elastic rod

wave equation has became one of the major issues widely concerned of Mathematics and Physics academia, because of its broad prospects for application in flexibility material mechanics. What’s more, the study of elastic rod wave equation has an important application value in revealing the law of wave propagation, explaining natural phenomena accurately, determining the physical material properties and so on. There are a lot of results for classical nonlinear elastic rod wave equation, Guopeng and Zhanglei put forward a class of nonlinear elastic rod wave equation, and obtained its approximate analytic solutions by the method of full approximation method. Tian Xiandong and Zhang Daheng discussed the travelling wave solutions of a new type of elastic rod wave equation (namely the nonlinear elastic rod wave equation). In recent years, there are a lot of methods to get

approximate analytic solutions of the nonlinear elastic rod wave equation, for example, the homogeneous balance method, F-expansion method, hyperbolic function expansion method, the trial function method, etc.

Now we consider the travelling wave solutions of the rod wave equation in the following form:

( )( ) (2 ), ( 0, )2t txx x x xx xxxg uu u u u uu t R (1)

where the function g( ) :u R R , R is a given constant. Eq.(1) is called generalized hyperelastic-rod wave equation. The existence of smooth traveling wave solutions for Eq.(1) is proved in [1].

In this paper, when ( )g is a quadratic polynomial, the traveling wave solutions of Eq.(1) is discussed. First the equation is transformed into the form of planar dynamic system by a series of transformations, using the bifurcation theory of planar dynamic system [2-5] and combining with the location of the balance point and its nature, its fork curve can be got, then topological phase portraits of the system can be drawn [6-8]. Through its first integral, traveling wave solutions of the implicit form, index form and triangle function form of the equation are obtained [9-10]. The remainder of the paper is organized as follows: In Section 2, phase portraits of the generalized hyperlastic-rod wave equation are given. In Section 3, traveling wave solutions of the implicit form, index form and triangle function form of the equation are obtained. The paper is concluded in Section 4.

II. PHASE PORTRAITS OF THE GENERALIZED HYPERLASTIC-ROD WAVE EQUATION

Set ( , ) ( )u t x x t ,( tx λξ −= ) is a smooth travelling wave solution of Eq.(1), then we have:

'( )' ''' ' (2 ' '' ''').2

g (2)

Integrate the Eq.(2) once about ξ ,we have 2( ) 1'' ( ') '' .

2 2gg (3)

where g is an integral constants. Set 'y , then Eq.(3) is equivalent to the following two-

dimensional system:

22 2 ( )

2( )

d yddy g g yd

(4)

which has the following first integral 3 2 2( , ) 2 (6 3 ) (2 ) 6( )H y a b g c y h (5)

where h is a constant. Considering that system (4) have a singular straight line

= , in order to avoid the singularity, make the

transform ( )d d , then system (4) becomes

2012 Fifth International Conference on Information and Computing Science

2160-7443/12 $26.00 © 2012 IEEE

DOI 10.1109/ICIC.2012.50

221

2012 Fifth International Conference on Information and Computing Science

2160-7443/12 $26.00 © 2012 IEEE

DOI 10.1109/ICIC.2012.50

221

2012 Fifth International Conference on Information and Computing Science

2160-7443/12 $26.00 © 2012 IEEE

DOI 10.1109/ICIC.2012.50

217

2

2( )

2 2 ( )

d yddy g g yd

(6)

System (4) and system (6) has the same first integral in the form of (5). Therefore, system (6) and system (4) have the same topological phase portraits in addition to the singular

line = . Clearly, = is a solution of system (6).

Giving a value, (5) decide a constant curve of system (6). As h changes, (5) determines different orbits of system (6) with different dynamic behavior. Here, we consider

0,0 >> γa and the case when ( )g is a quadratic polynomial, that is 2( )g a b c .For other cases, we can discuss them similarly.

Suppose ( , )e eM y is the coefficient matrix of system (6)

at the equilibrium point ( , )e ey , then 2 2( )

( , )'( ) 2 2

e ee e

e e

yM y

g y

and 2 2( , ) det ( , ) 4 2( )( '( ) 2 )e e e e e e eJ y M y y g ,

( , ) trace ( ( , )) 0e e e ey M yp = = .

According to the first integral (5) of system (6), and combined with the properties of its equilibrium points, we can get its bifurcation curves in the following form:

aacbg

84)2()(

2

1−−= λλ (7)

2

22

2 2)2()(

γγγλλγλ cbag −−−= (8)

For a given 0≠λ , Set

)(,2

)2(4)2(21

2

1 λλλφ gga

cgabb≤

+−−±−=±

(9)

)(,)2(2

22

32223

1 λγ

γλγλγγγgg

cbagy ≥

++−+=±

(10)

Specially, when )(1 λgg = , note −+ == 110 φφφ . Then we can

have the following properties: (1) when

)(2 ab

−=

γγλ , we have )()( 21 λλ gg = .

(a) If )(1 λgg < , then system (6) has two equilibrium points )0,( 1+φ and )0,( 1−φ .They are saddle points.

(b) If )(1 λgg = , then system (6) has two equilibrium points )0,( 0φ and )0,(

γλ .They are sharp points.

(c) If )(1 λgg > , then system (6) has two equilibrium points ),( 1+yγ

λ and ),( 1−yγλ .They are saddle points.

(2) when )(2 a

b−

≠γ

γλ , we have )()( 21 λλ gg > .

(a) If )(2 λgg < , then system(6) has two equilibrium points )0,( 1+φ and )0,( 1−φ .

(i) when )(2 a

b−

>γ

γλ , )0,( 1+φ and )0,( 1−φ are saddle points

for a>γ and a<γ respectively. (ii) when

)(2 ab

−<

γγλ , )0,( 1+φ and )0,( 1−φ are saddle points

for a<γ and a>γ respectively. (b) If )(2 λgg = , then system(6) has three equilibrium

points )0,( 1+φ , )0,( 1−φ and )0,(γλ . For equilibrium points )0,( 1+φ

and )0,( 1−φ , they are similar to that in (a), )0,(γλ is a sharp

point. (c) If )()( 12 λλ ggg << , then system (6) has four

equilibrium points )0,( 1+φ , )0,( 1−φ , ),( 1+yγλ and ),( 1−yγ

λ . For

equilibrium points )0,( 1+φ and )0,( 1−φ , they are similar to that in (a), ),( 1+yγ

λ and ),( 1−yγλ are saddle points.

(d) If )(1 λgg = , then system (6) has three equilibrium points )0,( 0φ , ),( 1+yγ

λ and ),( 1−yγλ . )0,( 0φ is a sharp point,

),( 1+yγλ and ),( 1−yγ

λ are saddle points.

(e) If )(1 λgg > , then system (6) has two equilibrium points ),( 1+yγ

λ and ),( 1−yγλ .They are saddle points.

Some phase portraits of system (6) are shown in Fig.1-Fig6.

Fig.1 Phase portrait of system (6) for

1, ( ).2( )b g ga

= <−

222222218

Fig.2 Phase portrait of system (6) for

2, ( ).2( )b g ga

= <−

Fig.3 Phase portrait of system (6) for

2, ( ).2( )b g ga

> <−

Fig.4 Phase portrait of system (6) for

2, ( ).2( )b g ga

> =−

Fig.5 Phase portrait of system (6) for

2, ( ).2( )b g ga

< =−

Fig.6 Phase portrait of system (6) for

2 1, ( ) ( ).2( )b g g ga

> < <−

III. TRAVELING WAVE SOLUTIONS OF THE EQUATION According to the first integral (5), we have

)(6)2()2(32 23

2

λγφφφλφ

−++−++= cgbahy

)(6

)()632()(22

2

λγφ

λγφφγ

λγ

λλγφγφ

−

−−++−=

baa

)(6

))(2)2(32( 23

2

λγφ

λγφγγ

λλγ

λ

−

−++−++

cgba

)(6

)2()2(322

2

3

3

λγφγ

λγ

λλγ

λ

−

+++−++

hcgba

when 0)2()2(322

2

3

3

=+++−+ hcgbaγ

λγ

λλγ

λ ,

223223219

Note γ3aA = ,

γλ

γλ

22

3 2

−+= baB

3

22

6)2()2(32

γγλγλλ cgbaC ++−+= .

then CBAy ++= φφ 22 (11)

that is

CBAdd ++±== φφφξφ 2' (12)

According to the indefinite integral formula Case1: when 0>A , we have solutions of the implicit form

ξφφφ ±=+++++ 0222ln1 CCBAABA

A (13)

where C0 is an integral constants. Especially, when 042 =− ACB and 0>A , (12) is reduced

to ξ

φφ dCA

d ±=+

(14)

So FEe A −±= ± ξφ . (15)

where E, F are constants. Now, take 2, 2, 2, 4, 3a b c γ λ= = = = = , 1, 0E F= = for

example, then 1 1 3, ,6 8 128

A B C= = = , we can give the profile

of the traveling wave solution as Fig.7.

Fig.7 The profile of the traveling wave solution 3

( , ) exp( ).6

x tu x t

−=

−

Case2: when 042 ≥− ACB and 0<A ,we have the solutions

of the following form

ξφ ±=+−−

− 0arcsin2 Caba

A (16)

That is

ABCA

AACB

2)(cos

24

0

2

−−±−−±= ξφ (17)

where C0 is an integral constants.

IV. CONCLUSIONS In this paper, a kind of generalized hyperlastic-rod wave

equation is studied. First the equation is transformed into the form of planar dynamic system by a series of transformations. Then the properties of equilibrium points and the orbits corresponding to them are studied by using the bifurcation theory of planar dynamic system. What’s more, topological phase portraits of the system are given. Through its first integral, traveling wave solutions of the implicit form, index form and triangle function form of the equation are obtained.

ACKNOWLEDGMENT This work was supported by the National Nature Science

foundation of China (Grant 70571030, 90610031), the Society Science Foundation from Ministry of Education of China (Grant 12YJAZH002, 08JA790057), A Project Funded by The Priority Academic Program Development of Jiangsu Higher Education Institutions, the Advanced Talents’ Foundation of Jiangsu University (No. 07JDG054), and the Student Research Foundation of Jiangsu University (No. 10A147).

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