cnoidal solutions, shock waves, and solitary … · pacs: 03.75.lm; 94.05.fg; 52.50.lp; 52.35.sb....

CNOIDAL SOLUTIONS, SHOCK WAVES, AND SOLITARY WAVESOLUTIONS OF THE IMPROVED KORTEWEG-DE VRIES EQUATION

P. SANCHEZ

Department of Mathematical Sciences,Delaware State University, Dover, DE 19901-2277, USA

E-mail: [email protected]

Received November 11, 2014

In this paper we obtain families of exact solutions of the improved Korteweg-de Vries equation with power law nonlinearity. Three integration schemes are applied.They are the travelling wave hypothesis, the ansatz approach, and the semi-inverse vari-ational principle. With the help of soliton perturbation theory, the adiabatic dynamicsof soliton parameters was also obtained.

Key words: Solitons; cnoidal waves; shock waves; integrability.

PACS: 03.75.Lm; 94.05.Fg; 52.50.Lp; 52.35.Sb.

1. INTRODUCTION

The dynamics of nonlinear waves in both classical and quantum physical sys-tems is one of the recurrent themes of special scientific interest [1-52]. A few ofthe models used to study shallow water waves along lake shores and beaches aregoverned by the Korteweg-de Vries (KdV) equation [4, 8], Boussinesq equation[20], Konopelchenko-Dubrovsky equation [21], Benjamin-Bona Mahoney equation[17] [also known as regularized long wave (RLW) equation or Peregrine equation],Rosenau-Kawahara equation [23], and Rosenau-KdV-RLW equation [34, 35], wherethe last two models typically describe dispersive shallow water waves. For two-layered fluid flows, some important models are Gear-Grimshaw equation [9, 37],Bona-Chen equation [10] and several others. This paper addresses the KdV equa-tion with an additional dispersion term that is of spatio-temporal type. This makessuch modified KdV equation an improved KdV model for addressing shallow waterdynamics [22].

The integrability aspects of the improved KdV equation with power law nonli-nearity will be studied in this paper. Three kinds of integration tools will be applied:traveling wave hypothesis, ansatz approach, and He’s semi-inverse variational princi-ple (SVP). By using the traveling wave hypothesis we will obtain cnoidal-wave solu-tions and, in the limiting case, the corresponding solitary waves. The ansatz approachwill reveal solitary waves, shock waves, and finally singular solitary waves that modelrogue waves. There are several integrability conditions, also known as constraint con-ditions, which will be revealed during the analysis of the improved KdV equation.

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380 P. Sanchez 2

Then, He’s SVP will be applied to retrieve the corresponding solitary-wave solu-tions. Subsequently, the conservation laws will be revisited in this paper. Finally,the soliton perturbation theory will display the adiabatic dynamics of solitary-waveparameters.

2. TRAVELLING WAVE SOLUTIONS

The improved KdV equation is given by [22, 31]:

qt+aqnqx+ b1qxxx+ b2qxxt = 0. (1)

In (1), the dependent variable q(x,t) represents the wave profile with x and t beingthe independent variables. Here x is the spatial variable and t is the temporal variable.The two dispersion terms in eq. (1) are indicated by the coefficients bj for j = 1,2.The inclusion of the b2 term makes this equation an improved KdV equation. In thissection we will apply the traveling wave hypothesis to retrieve cnoidal waves andhence in the limiting case, solitary wave solutions to the problem. The parameter nindicates the exponent of the power law nonlinearity. The starting hypothesis is

q(x,t) = g(x−vt) = g(s), (2)

where

s= x−vt (3)

We substitute the above relation into eq. (1) to get:

−vg′+agng′+ (b1− b2v)g′′′ = 0. (4)

We then integrate once:

−vg+a

n+ 1gn+1 + (b1− b2v)g′′ =K1, (5)

where K1 is an integration constant.We multiply by g′ and integrate again:

−v2g2 +

a

(n+ 1)(n+ 2)gn+2 +

b1− b2v2

(g′)2 =K1g+K2, (6)

where K2 is a second integration constant.Travelling wave solutions will only exist for n= 1. Substituting this value for

n and introducing a function G(g) leads to:

(g′)2 =− a

3(b1− b2v)g3 +

v

b1− b2vg2 +K1g+K2 =G(g). (7)

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2.1. CNOIDAL WAVES

First, we consider the case with G(g) in eq. (7) having three real distinct rootssuch that g3 < g2 < g1. Then G(g) can be written as:

G(g) =− a

3(b1− b2v)(g−g1)(g−g2)(g−g3) (8)

s= s3±∫ g

g3

df[− a

3(b1−b2v)(f −g1)(f −g2)(f −g3)] 1

2

. (9)

To get a cnoidal wave solution, we use the following substitution:

f = g3 + (g2−g3)sin2 θ. (10)

Then

s= s3±2

√3(b1− b2v)

a(g3−g1)

∫ ψ

0

dψ√1−msin2ψ

, (11)

wherem=

g2−g3g1−g3

, (12)

which in this case represents the modulus of the corresponding eliptic function. Theorder of the distinct roots guarantee that 0<m< 1. Hence, the cnoidal-wave solutionis given by:

g(s) = g2− (g2−g3)cn2[s−s3

2

√a(g3−g1)3(b1− b2v)

|m

]. (13)

2.2. SOLITARY WAVE SOLUTIONS

To get solitary wave solutions, we introduce boundary conditions as follows:

g, g′, g′′→ 0 as |s| →∞, (14)

which leads to K1 =K2 = 0. Hence, eq. (7) becomes:

g′ = g

√v

b1− b2v− a

3(b1− b2v)g, (15)

whose solution is ∫dg

g√

vb1−b2v −

a3(b1−b2v)g

=±∫ds. (16)

Then the solitary wave solution is given by:

g(x−vt) =3v

asech2

[1

2

√v

b1− b2v(x−vt−x0)

]. (17)

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3. ANSATZ APPROACH

The second method of integrability is the ansatz approach. There are someminor disadvantages with traveling wave hypothesis that was studied in the previ-ous section. The speed of the solitary wave was not recovered from the analysis.However, it is important to retrieve the value of that parameter in order to get a com-plete spectrum of soliton parameters. The ansatz approach enables to circumventthis drawback. Another hindrance of the previous approach is that it is limited to thedetermination of solitary waves. Shock waves and singular soliton solutions are notrecovered by the traveling wave approach.

Therefore, the ansatz approach that was proposed by Biswas et al. [7, 8, 11,12] is a direct method to retrieving soliton solutions to nonlinear evolution equations(NLEEs). Consequently, the popularity of this simple scheme exploded within a veryshort time frame. In fact, this integration algorithm is applicable to coupled nonlinearevolution systems as well. Additionally, this scheme can be used to study NLEEs thatcarry time-dependent coefficients.

This method starts with a guess for the soliton solution, depending on the typeof soliton or shock wave. This guess is substituted into the NLEE that is understudy. The remaining set of algebraic equations leads to the exact solution withcorresponding parameter dependencies as well as the speed of the nonlinear wave.

3.1. SOLITARY WAVES

For solitary waves, the starting hypothesis is [31]

q =Asechp [B(x−vt)] =Asechpτ, (18)

where (A) is the soliton amplitude, (B) is the inverse width, and (v) is the velocityof the wave. The unknown exponent p will be determined in terms of the power lawnonlinearity parameter n. Substituting (18) into (1) leads to:[

v− b1p2B2 + b2vp2B2

]sechpτ −aAnsechp(n+1)τ

+ (b1− b2v)(p+ 1)(p+ 2)B2sechp+2τ = 0.(19)

By the balancing principle, we equate the exponents p(n+1) and p+2, which givesthe relation between the unknown parameter p and the power law nonlinearity para-meter n

p=2

n. (20)

The solitary wave solution is given by

q(x,t) =Asech2n [B(x−vt)] . (21)

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The relation between the soliton amplitude (A) and the inverse width (B) is given by

A=

[2(n+ 1)(n+ 2)(b1− b2v)B2

an2

] 1n

, (22)

and the velocity (v) of the soliton is

v =4b1B

2

n2 + 4b2B2. (23)

We substitute this expression for the velocity into equation (22) to get the relationbetween amplitude and width independent from the velocity:

A=

[2(n+ 1)(n+ 2)b1B

2

a(n2 + 4b2B2)

] 1n

. (24)

It needs to be noted that this section 3.1 is a revisitation of the results that wereobtained earlier [31]. Nevertheless, the above results are just included here in orderto paint a complete picture of the model. The details of the derivation are howeveromitted since they were reported earlier [31].

3.2. SINGULAR SOLITONS

Here the starting hypothesis is taken to be:

q =Acschp [B(x−vt)] =Acschpτ, (25)

where (A) and (B) are free parameters, and (v) is the velocity of the wave. Theunknown exponent p will be determined in terms of the power law nonlinearity pa-rameter n. Substituting (25) into (1) leads to:[

v− b1p2B2 + b2vp2B2

]cschpτ −aAncschp(n+1)τ

+ (b1− b2v)(p+ 1)(p+ 2)B2cschp+2τ = 0.(26)

By the balancing principle, we equate the exponents p(n+1) and p+2, which givesthe relation between unknown parameter p and power law nonlinearity parameter n

p=2

n. (27)

The singular wave solution is

q(x,t) =Acsch2n [B(x−vt)] . (28)

The relation between the free parameters (A) and (B) is given by

A=

[2(n+ 1)(n+ 2)(b1− b2v)B2

an2

] 1n

, (29)

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384 P. Sanchez 6

and the velocity (v) of the soliton is

v =4b1B

2

n2 + 4b2B2. (30)

We substitute this expression for the velocity into equation (29) to get the relationbetween the free parameters that is independent from the velocity:

A=

[2(n+ 1)(n+ 2)b1B

2

a(n2 + 4b2B2)

] 1n

. (31)

3.3. SHOCK WAVES

The starting hypothesis is

q =Atanhp [B(x−vt)] =Atanhp τ, (32)

where (A) and (B) are free parameters, and (v) is the velocity of the wave. Theunknown exponent p will be determined in terms of the power law nonlinearity pa-rameter n. Substituting (32) into (1) leads to:[

v+ (b1− b2v)(3p2 + 3p+ 2)B2]

tanhp+1τ

−[v+ (b1− b2v)(3p2−3p+ 2)B2

]tanhp−1τ

+aAntanhp(n+1)−1τ −aAntanhp(n+1)+1τ

−[(b1− b2v)(p+ 1)(p+ 2)B2

]tanhp+3τ

+[(b1− b2v)(p−1)(p−2)B2

]tanhp−3τ = 0.

(33)

By the balancing principle, we equate the exponents p(n+1) and p+2, which givesthe relation between the unknown parameter p and the power law nonlinearity para-meter n. We substitute the ansatz into (15), and by the balancing principle we get

np+p+ 1 = p+ 3 (34)

np+p−1 = p+ 1 (35)

p=2

n(36)

From the coefficient of tanhp−3 τ we obtain

(b1− b2v)AB3p(p−1)(p−2) = 0. (37)

This leads to two main cases.

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3.3.1. CASE-I: p= 1, n= 2

The topological soliton solution of (1) is given by:

q =Atanh[B(x−vt)] , (38)

where the velocity of the soliton is given by:

v =− 2b1B2

1−2b2B2(39)

The relation between the free parameters A and B is given by:

A=

[−v+ 8(b1− b2v)B2

a

] 12

(40)

or

A=

[−6(b1− b2v)B2

a

] 12

. (41)

These relations prompt the respective constraints given by

a{v+ 8(b1− b2v)B2

}< 0 (42)

anda{

(b1− b2v)B2}< 0. (43)

Upon equating eq. (40) and eq. (41) we can extract the same expression for thevelocity as in (39).

3.3.2. CASE-II: p= 2, n= 1


q =Atanh2 [B(x−vt)] , (44)

where the velocity of the soliton is given by:

v =− 8b1B2

1−8b2B2. (45)


A=−v+ 20(b1− b2v)B2

a(46)

or

A=−12(b1− b2v)B2

a. (47)

Upon equating these relations we can extract the same expression for the velocity asin (45).

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4. CONSERVATION LAWS

The three conserved quantities that (1) possesses are given by mass (M ), mo-mentum (P ), and energy (E) [20]

M =

∫ ∞−∞

qdx=A

B

Γ(12

)Γ(1n

)Γ(12 + 1

n

) , (48)

P =

∫ ∞−∞

(q2− b2q2x

)dx

=A2

B

Γ(12

)Γ(2n

)Γ(12 + 2

n

) [n(n+ 4)−4b2B2]

n(n+ 4),

(49)

and

E =

∫ ∞−∞

{2aqn+2

(n+ 1)(n+ 2)− b1q2x

}dx

=aAn [b1(n+ 1)(n+ 2)(4−n)−8b2A

n]

(n+ 1)(n+ 2)(n+ 4)[2(n+ 1)(n+ 2)b1−4b2An].

(50)

If the second dispersion term with b2 is not present in the improved KdV equation(1), the expression for energy shows that the results are consistent with the ones thatwere reported earlier [4, 15]. Therefore, for b2 = 0, one can again conclude that theKdV equation with power law nonlinearity exists provided n 6= 4.

5. SEMI-INVERSE VARIATIONAL PRINCIPLE

This is the third approach to the integrability aspect of the improved KdV equa-tion. The SVP was first proposed more than a decade ago by He and this method hasgained popularity ever since its first appearance [11, 23, 33, 34, 35].

One starts with the traveling wave hypothesis to (1) as

q(x,t) = g(x−vt) = g(s), (51)

wheres= x−vt. (52)

Substituting this traveling wave assumption into (1) and integrating once while takingthe integration constant to be zero gives

−vg+a

n+ 1gn+1 + (b1− b2v)g′′ = 0 (53)

Now, multiplying both sides of (53) by g′ and integrating leads to

−v2g2 +

a

(n+ 1)(n+ 2)gn+2 +

b1− b2v2

(g′)2

=K, (54)

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where K is the integration constant. The stationary integral is then defined as

J =

∫ ∞−∞

Kds

=

∫ ∞−∞

[−v

2g2 +

a

(n+ 1)(n+ 2)gn+2 +

b1− b2v2

(g′)2]

ds.

(55)

Now, choosing

g(s) =Asech2n (Bs) (56)

as a solution hypothesis for (1), where A and B are still the amplitude and inversewidth of the soliton, the stationary integral J reduces to

J =A2

B

Γ(12

)Γ(2n

)Γ(12 + 2

n

) ××{−v

2+

2(b1− b2v)B2

n(n+ 4)+

4aAn

(n+ 1)(n+ 2)(n+ 4)

}.

(57)

Then, the SVP states that the amplitude (A) and the width (B) are given by thesolution of the coupled system of equations

∂J

∂A= 0 (58)

and∂J

∂B= 0 (59)

From (57), the equations (58) and (59) are respectively given by

4(n+ 1)(b1− b2v)B2−n(n+ 1)(n+ 4)v+ 4anAn = 0 (60)

and

4(n+ 1)(n+ 2)(b1− b2v)B2 +n(n+ 1)(n+ 2)(n+ 4)v−8anAn = 0. (61)

We then eliminate An from the above equations to get an expression for the inversewidth B:

B =n

2

√− v

(b1− b2v)(62)

with the constraint condition

v(b1− b2v)< 0. (63)

Finally, we eliminate B from the coupled equations to get the amplitude A:

A=

[(n+ 1)(n+ 2)v

2

] 1n

. (64)

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6. SOLITON PERTURBATION THEORY

The dynamics of shallow water waves always comes with perturbation terms.The rest of the paper will therefore be devoted to the perturbed improved KdV equa-tion. This section will study the adiabatic dynamics of soliton parameters when per-turbation terms are turned on.

6.1. MODIFIED CONSERVATION LAWS

In the presence of perturbation terms, the improved KdV equation is given by[33, 35]

qt+aqnqx+ b1qxxx+ b2qxxt = εR, (65)where ε represents the perturbation parameter and R represents the perturbationterms. Here, it is tacitly assumed that 0 < ε� 1. In the presence of perturbationterms, the conservation laws are modified because these conserved quantities un-dergo adiabatic deformation. Therefore, these modified conservation laws are givenby

dM

dt= ε

∫ ∞−∞

Rdx (66)

dP

dt= 2ε

∫ ∞−∞

qRdx (67)

dE

dt= 2ε

∫ ∞−∞

[aqn+1

n+ 1R− (b1qx+ b2qt)

∂

∂xR

], (68)

which respectively represent the adiabatic variation of the soliton mass, the solitonmomentum, and the soliton energy. The slow change in the soliton velocity is givenby

v =4b1B

2

n2 + 4b2B2+

ε

M

∫ ∞−∞

xRdx. (69)

6.2. ADIABATIC PARAMETER DYNAMICS

In this paper, the specific perturbation terms that will be taken into considera-tion are given by [33, 35]

R= αq+βqxx+γqxqxx+ δqmqx+λqqxxx+νqqxqxx

+σq3x+ ξqxqxxxx+ηqxxqxxx+ρqxxxx+ψqxxxxx+κqqxxxxx(70)

To describe perturbation effects: α - term represents the shoaling, β is the dissipationcoefficient, ψ is the fifth order spatial dispersion coefficient, and ρ -term is a higherorder stabilization. The higher order nonlinear dispersion term is represented by the

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δ-term, where 1 ≤ m ≤ 4. The rest of the perturbation terms are described in theWhitham hierarchy.So, the perturbed equation considered in this paper is

qt+aqnqx+ b1qxxx+ b2qxxt

= ε(αq+βqxx+γqxqxx+ δqmqx+λqqxxx+νqqxqxx

+ σq3x+ ξqxqxxxx+ηqxxqxxx+ρqxxxx+ψqxxxxx+κqqxxxxx).

(71)

In the presence of these perturbation terms, the adiabatic change of the mass is givenby

dM

dt=εαA

B

Γ(12

)Γ(1n

)Γ(12 + 1

n

) = εαM, (72)

which shows thatM(t) =M0e

εαt, (73)where M0 is the initial mass of the soliton. Hence, in the limiting case

limt→∞

M(t) = 0, (74)

for α< 0. This shows that the mass adiabatically dissipates with time, in the presenceof shoaling since this is a dissipative perturbation term. The adiabatic change of themomentum is given by

dP

dt=

2εA2

Bn2(n+ 4)(3n+ 4)

Γ(12

)Γ(2n

)Γ(12 + 2

n

) ××{α(3n+ 4)(n+ 4)n2−4β(3n+ 4)nB2−16ρ(2n+ 3)B4

}.

(75)

This means that when dP/dt = 0, the solitons will travel with constant momentumfor a stable fixed value of the width given by

B =

[−βn(3n+ 4)

8ρ(2n+ 3)±n√

(3n+ 4)[β2(3n+ 4) + 4ρα(2n+ 3)(n+ 4)]

8ρ(2n+ 3)

] 12

, (76)

with the constraint condition

β2(3n+ 4)≥−4ρα(2n+ 3)(n+ 4). (77)

The corresponding fixed value of the amplitude, in this case by virtue of (24), is

A=

b1(n+ 1)(n+ 2)[−(3n+ 4)β±

√(3n+ 4)D1

]2a{

2ρn(2n+ 3) + b2

[−(3n+ 4)β±

√(3n+ 4)D1

]}

1n

, (78)

whereD1 = β2(3n+ 4) + 4ρα(2n+ 3)(n+ 4). (79)

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390 P. Sanchez 12

The adiabatic change of the energy is given by

dE

dt=

8εA2

Bn4(5n+ 4)(3n+ 4)(n+ 4)

Γ(12

)Γ(2n

)Γ(12 + 2

n

) ××{aAnD2

(n+ 1)− (b1− b2v)B2D3

n2

},

(80)

where

D2 = αn4(3n+ 4)(5n+ 4)−4βn3(5n+ 4)(n+ 1)B2

+ 16ρn2(n+ 1)(n2 + 5n+ 3)B4(81)

and

D3 = αn5(3n+ 4)(5n+ 4)−4βn4(5n+ 4)(2n+ 3)B2

−16ρ(26n6 + 188n5 + 518n4 + 715n3 + 536n2 + 208n+ 32)B4.(82)

This means that when dE/dt = 0, the solitons will travel with constant energy forthe given relation between amplitude and inverse width

A=

[B2(b1− b2v)(n+ 1)D4

an2D3

] 1n

. (83)

For these specific perturbation terms, given by (70), the slow change in the velocityof the soliton is

v =4b1B

2

n2 + 4b2B2− εAmB

(m+ 1)

Γ(12 + 1

n

)Γ(m+1n

)Γ(m+1n + 1

2

)Γ(1n

)− 4εA2B3 [nν+ (n+ 9)σ]

3n(n+ 3)(n+ 6)

Γ(12 + 1

n

)Γ(3n

)Γ(3n + 1

2

)Γ(1n

)+

2εAB3

n(n+ 4)

{−γ+ 3λ+

4B2(2n+ 3)[3ξ−η−5κ]

n(3n+ 4)

}Γ(12 + 1

n

)Γ(2n

)Γ(2n + 1

2

)Γ(1n

) ,(84)

by virtue of (69).

7. EXACT SOLITARY WAVE SOLUTION

The perturbed improved KdV equation that will be considered in this paper is[33, 35]

qt+aqnqx+ b1qxxx+ b2qxxt =R, (85)where the perturbation terms are given by

R= γqxqxx+λqqxxx+νqqxqxx+σq3x

+ ξqxqxxxx+ηqxxqxxx+κqqxxxxx,(86)

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and the parameter ε is set to 1 so that the perturbation parameters are treated in astrong sense. The starting hypothesis is:

q =Asechp [B(x−vt)] =Asechpτ. (87)

We substitute this ansatz into (85):[v− b1p2B2 + b2vp

2B2]

sechpτ −aAnsechp(n+1)τ

+ (b1− b2v)(p+ 1)(p+ 2)B2sechp+2τ

=−p2AB2[γ+λ+p2B2(ξ+η+κ)

]sech2pτ

+ (p+ 1)AB2[γp+λ(p+ 2)

+ 2{ηp2(p+ 1) + (p2 + 2p+ 2)[ξp+κ(p+ 2)]

}B2]sech2(p+1)τ

− (p+ 1)(p+ 2)AB4 [ξp(p+ 3) +ηp(p+ 1) +κ(p+ 3)(p+ 4)]sech2(p+2)τ

−p2A2B2(ν+σ)sech3pτ +pA2B2 [ν(p+ 1) +σp]sech3p+2τ

(88)

By the balancing principle, we equate the exponents p(n+1) and p+2, which leadsto

p=2

n. (89)

The solitary wave solution is given by

q(x,t) =Asech2n [B(x−vt)] . (90)

The relation between the soliton amplitude A and the inverse width B is given by

A=

[2(n+ 1)(n+ 2)(b1− b2v)B2

an2

] 1n

, (91)

and the velocity v of the soliton is

v =4b1B

2

n2 + 4b2B2. (92)

We substitute this expression for the velocity into equation (91) to get the relationbetween amplitude and width independent from the velocity:

A=

[2(n+ 1)(n+ 2)b1B

2

a(n2 + 4b2B2)

] 1n

. (93)

The additional linearly independent functions lead to a set of constraint relationsgiven by

σ+ν = 0 (94)

2σ+ (n+ 2)ν = 0 (95)

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392 P. Sanchez 14

This linear system of equations implies the unique solution[σν

]=

[00

]. (96)

The last set of constraint conditions is given by:

n2(γ+λ) + 4(ξ+η+κ)B2 = 0 (97)

n2(γ+ (n+ 1)λ) + 4{

(n2 + 2n+ 2)[ξ+ (n+ 1)κ] + (n+ 2)η}B2 = 0 (98)

(3n+ 2)ξ+ (n+ 2)η+ (3n+ 2)(2n+ 1)κ= 0. (99)This set of linear equations, given by (97)-(99) gives the solution

γλξηκ

=

−4(n+3)B2

n(3n+2)4(n+1)B2

n(3n+2)

− (n+2)(3n+2)

10

η+

−4B2

4(n+2)B2

n−(2n+ 1)

01

κ (100)

Thus, the system of linear equations (97)-(99) for the perturbation coefficients lead tothe conclusion that the five perturbation coefficients, ξ, η, κ, γ, and λ can be solvedin terms of two linearly independent parameters, namely η and κ only.

8. EXACT SINGULAR SOLITON SOLUTION

The starting hypothesis for the exact singular soliton solution is [35]:

q =Acschp [B(x−vt)] =Acschpτ. (101)

Substituting this ansatz into equation (85):[v− b1p2B2 + b2vp

2B2]

cschpτ −aAncschp(n+1)τ

+ (b1− b2v)(p+ 1)(p+ 2)B2cschp+2τ

=−p2AB2[γ+λ+p2B2(ξ+η+κ)

]csch2pτ

+ (p+ 1)AB2[γp+λ(p+ 2)

+ 2{ηp2(p+ 1) + (p2 + 2p+ 2)[ξp+κ(p+ 2)]

}B2]csch2(p+1)τ

− (p+ 1)(p+ 2)AB4 [ξp(p+ 3) +ηp(p+ 1) +κ(p+ 3)(p+ 4)]csch2(p+2)τ

−p2A2B2(ν+σ)csch3pτ +pA2B2 [ν(p+ 1) +σp]csch3p+2τ

(102)

RJP 60(Nos. 3-4), 379–400 (2015) (c) 2015 - v.1.3a*2015.4.21


Then balancing the exponents p(n+ 1) and p+ 2 leads to:

p=2

n. (103)

Finally, the singular soliton solution is given by:

q(x,t) =Acsch2n [B(x−vt)] . (104)

Note that the same expressions for the free parameter A and velocity v, which aregiven by (91)-(93), also apply for the singular soliton solution. The constraint condi-tions given by (94)-(100) for the perturbation terms considered for this problem holdas well.

9. EXACT SHOCK WAVE SOLUTION

The starting hypothesis for topological soliton solutions is as follows:

q =Atanhp [B(x−vt)] =Atanhp τ. (105)

We substitute this ansatz into (85):[v+ (b1− b2v)(3p2 + 3p+ 2)B2

]tanhp+1τ

−[v+ (b1− b2v)(3p2−3p+ 2)B2

]tanhp−1τ

+aAntanhp(n+1)−1τ −aAntanhp(n+1)+1τ

−[(b1− b2v)(p+ 1)(p+ 2)B2

]tanhp+3τ

+[(b1− b2v)(p−1)(p−2)B2

]tanhp−3τ

= (p−1)(p−2)AB4 [ξp(p−3) +ηp(p−1) +κ(p−3)(p−4)] tanh2p−5τ

− (p+ 1)(p+ 2)AB4 [ξp(p+ 3) +ηp(p+ 1) +κ(p+ 3)(p+ 4)] tanh2p+5τ

− (p+ 1)AB2[γp+λ(p+ 2)− ξp(5p2 + 13p+ 14)B2

− ηp(p+ 1)(5p+ 2)B2−5κ(p+ 2)(p2 + 3p+ 4)B2]

tanh2p+3τ

+ (p−1)AB2[γp+λ(p−2)− ξp(5p2−13p+ 14)B2

− ηp(p−1)(5p−2)B2−5κ(p−2)(p2−3p+ 4)B2]

tanh2p−3τ

−AB2[γp(3p−1) +λ(3p2−3p+ 2)−2ξp(5p3−6p2 + 13p−4)B2

− 2ηp2(5p2−4p+ 1)B2−2κ(5p4−10p3 + 25p2−20p+ 8)B2]

tanh2p−1τ

+AB2[γp(3p+ 1) +λ(3p2 + 3p+ 2)−2ξp(5p3 + 6p2 + 13p+ 4)B2

− 2ηp2(5p2 + 4p+ 1)B2−2κ(5p4 + 10p3 + 25p2 + 20p+ 8)B2]

tanh2p+1τ.

(106)

By the balancing principle we get

np+p+ 1 = p+ 3 (107)

RJP 60(Nos. 3-4), 379–400 (2015) (c) 2015 - v.1.3a*2015.4.21

394 P. Sanchez 16

np+p−1 = p+ 1 (108)

p=2

n(109)

From the coefficient of tanhp−3 τ

(b1− b2v)AB3p(p−1)(p−2) = 0 (110)

we can get two main cases.

9.1. CASE-I: p= 1, n= 2


q =Atanh[B(x−vt)] . (111)

The velocity of the soliton is given by:

v =− 2b1B2

1−2b2B2. (112)


A=

[−v+ 8(b1− b2v)B2

a

] 12

(113)

or

A=

[−6(b1− b2v)B2

a

] 12

(114)

These relations prompt the respective constraints given by

a{v+ 8(b1− b2v)B2

}< 0 (115)

anda{

(b1− b2v)B2}< 0 (116)

Upon equating eq. (113) and eq. (114) we can extract the same expression for thevelocity as in (112).The additional linearly independent functions lead to a set of constraint relationsgiven by:

σ = 0 (117)

2ν+ 3σ = 0 (118)

4ν+ 3σ = 0 (119)

RJP 60(Nos. 3-4), 379–400 (2015) (c) 2015 - v.1.3a*2015.4.21


2ν+σ = 0 (120)

γ+λ−2B2(4ξ+η+ 4κ) = 0 (121)

γ+ 2λ−B2(14ξ+ 5η+ 34κ) = 0 (122)

γ+ 3λ−2B2(16ξ+ 7η+ 60κ) = 0 (123)

2ξ+η+ 10κ= 0. (124)Equations (117)-(120) lead to a unique solution:[

σν

]=

[00

]. (125)

The system of linear equations (121)-(124) for the five perturbation coefficients, ξ,η, κ, γ, and λ can be solved in terms of only two linearly independent parameters ηand κ.

γλξηκ

=

−2B2

0−1

210

η+

−28B2

−4B2

−501

κ (126)

9.2. CASE-II: p= 2, n= 1


q =Atanh2 [B(x−vt)] . (127)

The velocity of the soliton is given by:

v =−8b1B

2

1−8b2B2(128)


A=−v+ 20(b1− b2v)B2

a(129)

or

A=−12(b1− b2v)B2

a. (130)

Upon equating these expressions we can extract the same expression for the velocityas in (128).

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396 P. Sanchez 18

The additional linearly independent functions lead to a set of constraint relationsgiven by:

ν+σ = 0 (131)

5ν+ 6σ = 0 (132)

7ν+ 6σ = 0 (133)

3ν+ 2σ = 0 (134)

γ−2B2(4ξ+ 9η) = 0 (135)

5γ+ 4λ−2B2(38ξ+ 26η+ 17κ) = 0 (136)

7γ+ 10λ−2B2(94ξ+ 29η+ 77κ) = 0 (137)

γ+ 2λ−4B2(15ξ+ 9η+ 35κ) = 0 (138)

5ξ+ 3η+ 15κ= 0. (139)Equations (131)-(134) lead to a unique solution:[

σν

]=

[00

]. (140)

The system of linear equations (135)-(139) for the five perturbation coefficients, ξ,η, κ, γ and λ can be solved for a unique solution:

γλξηκ

=

00000

(141)

10. THE SEMI-INVERSE VARIATIONAL PRINCIPLE (THE PERTURBED CASE)

This section will focus on the integrability aspect of the perturbed improvedKdV equation, where the perturbation terms are all of Hamiltonian type in order topermit integrability:

qt+aqx+ bqxxx+ cqxxxxt+k (qn)x = δqmqx+ψqxxxxx. (142)

RJP 60(Nos. 3-4), 379–400 (2015) (c) 2015 - v.1.3a*2015.4.21


The traveling wave hypothesis is taken to be

q(x,t) = g(x−vt) = g(s), (143)

wheres= x−vt. (144)

Substituting this traveling wave assumption into (142) and integrating once whiletaking the integration constant to be zero gives

−vg+a

n+ 1gn+1 +

δ

m+ 1gm+1 + (b1− b2v)g′′+ψg′′′′ = 0. (145)

Now, multiplying both sides of (145) by g′ and integrating leads to

− v2g2 +

a

(n+ 1)(n+ 2)gn+2 +

b1− b2v2

(g′)2

+δ

(m+ 1)(m+ 2)gm+2 +ψ

[g′g′′′− 1

2(g′′)2

]=K,

(146)

where K is the integration constant. The stationary integral is then defined as

J =

∫ ∞−∞

Kds

=

∫ ∞−∞

[−v

2g2 +

a

(n+ 1)(n+ 2)gn+2 +

b1− b2v2

(g′)2

+δ

(m+ 1)(m+ 2)gm+2 +ψ

[g′g′′′− 1

2(g′′)2

]]ds.

(147)

Now, choosing

g(s) =Asech2n (Bs) (148)

as a solution hypothesis for (142), where A and B are still the amplitude and inversewidth of the soliton, the stationary integral J reduces to

J =A2

2B

Γ(12

)Γ(2n

)Γ(12 + 2

n

) { 8aAn

(n+ 1)(n+ 2)(n+ 4)−v+

4(b1− b2v)B2

(n+ 4)n

+16ψ(2n+ 3)B4

(n+ 4)(3n+ 4)n2

}+

δAm+2

(m+ 1)(m+ 2)B

Γ(12

)Γ(m+2n

)Γ(12 + m+2

n

) . (149)

Then SVP gives

∂J

∂A=A

B

Γ(12

)Γ(2n

)Γ(12 + 2

n

) { 4aAn

(n+ 1)(n+ 4)−v+

4(b1− b2v)B2

(n+ 4)n

+16ψ(2n+ 3)B4

(n+ 4)(3n+ 4)n2

}+

δAm+1

(m+ 1)B

Γ(12

)Γ(m+2n

)Γ(12 + m+2

n

) = 0

(150)

RJP 60(Nos. 3-4), 379–400 (2015) (c) 2015 - v.1.3a*2015.4.21

398 P. Sanchez 20

and

∂J

∂B=A2

2

Γ(12

)Γ(2n

)Γ(12 + 2

n

) { −8aAn

(n+ 1)(n+ 2)(n+ 4)B2+

v

B2+

4(b1− b2v)

(n+ 4)n

+48ψ(2n+ 3)B2

(n+ 4)(3n+ 4)n2

}− δAm+2

(m+ 1)(m+ 2)B2

Γ(12

)Γ(m+2n

)Γ(12 + m+2

n

) = 0,

(151)

which leads to the amplitude-width relationship given by

A=

[− (n+ 1)(n+ 2)

8a(3n+ 4)n2(n−m)×

×{mv(n+ 4)(3n+ 4)n2 + 4(b1− b2v)(3n+ 4)n(m+ 4)B2

+ 16ψ(2n+ 3)(3m+ 8)B4}] 1

m .

(152)

11. CONCLUSIONS

In this paper we have obtained exact solitary-wave, shock-wave, and cnoidal-wave solutions of the improved Korteweg-de Vries equation with power law nonli-nearity. Additionally, the soliton perturbation theory recovered adiabatic dynamicsof soliton parameters. These results will be further generalized. The improvedKorteweg-de Vries equation with time-dependent coefficients will be studied. Thestochastic perturbation terms will be applied and the dynamics of soliton parameters,in the presence of random perturbations will be investigated. Those new results willbe reported elsewhere.

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