Exact Solution and Conservation Laws for Fifth Order Korteweg de Vries Equation

http://dx.doi.org/10.4236/jamp.2013.15007

Exact Solution and Conservation Laws for Fifth-Order

Korteweg-de Vries Equation

Elham M. Al-Ali

Mathematics Department, Faculty of Science, University of Tabuk, Tabuk, KSA Email: [email protected]

Received September 11, 2013; revised October 11, 2013; accepted October 17, 2013

Copyright © 2013 Elham M. Al-Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

With the aid of Mathematica, new exact travelling wave solutions for fifth-order KdV equation are obtained by using the solitary wave ansatz method and the Wu elimination method. The derivation of conservation laws for a fifth-order KdV equation is considered.

Keywords: Soliton Solutions; Conservation Laws; Nonlinear Evolution Equations

1. Introduction

It is well-known that nonlinear complex physical phe-nomena are related to nonlinear partial differential equa-tions (NLPDEs) which are involved in many fields from physics to biology, chemistry, mechanics, etc. As mathe- matical models of the phenomena, the investigation of exact solutions to the NLPDEs reveals to be very important for the understanding of these physical problems. Many mathematicians and physicists have well understood this importance when the importance of this so they decided to pay special attention to the development of sophisticated methods for constructing exact solutions to the NLPDEs. Thus, a number of powerful methods have been presented.

We can cite the inverse scattering transform [1], the Bäcklund and Darboux transform [2-5], Hirota’s bilinear method [6], the homogeneous balance method [7], Jacobi elliptic function method [8], the tanh-method and ex-tended tanh-function method [9-15], F-expansion method [16-18] and so on.

The notion of conservation laws is important in the study of nonlinear evolution equations (NLEEs) appear-ing in mathematical physics [19]. The mathematical ori-gin of conservation laws results from the formulation of familiar physical laws such as for mass, energy and mo-mentum [20]. As is known, the investigation of conser-vation laws of the Korteweg-de Vries (KdV) equation led to the discovery of a number of techniques to solve NLEEs [21], e.g., Miura transformation, Lax pair, in-verse scattering technique and bi-Hamiltonian structures.

On the other hand, it is useful in the numerical integra-tion of NLEEs [22] (e.g., to control numerical errors); particularly with regard to integrability and linearization, constants of motion, analysis of solutions, and numerical solution methods [23]. Consider a dynamical system,



, , , , , ,



t x t

u  f x t u u u u_xt  (1)

where uu x t

 

, is a function of two independent

variables t and x. The functional I u x t

 

,  is said to

be a constant of motion or an integral of Equation (1), if it satisfies d

 

,

dtI u x t 0. Generally, we can derive

constants of motion from conservation law, which enjoys the general form as [24]

 

 

d d

, ,

dxV u x t dtG u x t 0 (2)

while the components V and of the conserved

vector

G



V G,



are functions of x t, and derivatives of

. The equality (2) is assumed to be satisfied for any solution of the corresponding system of equations, is called conserved density and G is called conserved

flow. With the assumption that the function

u

V

 

,t u x and its derivatives with respect to x go to zero sufficiently

fast as x  

 

,

 

, d

I u x t V x t x 



   

 



  (3)

number of conservation laws such as the fifth-order KdV equation

2. Exact Solution for Fifth-Order KdV

Equation

With the rapid development of science and technology, the study kernel of modern science is changed from lin-ear to nonlinlin-ear step by step. Many nonlinlin-ear science problems can simply and exactly be described by using the mathematical model of nonlinear equation. Up to now, many important physical nonlinear evolution tions are found, such as sin-Gordon equation, KdV equa-tions, Schrodinger equation all possess solitary wave solutions. There exist many methods to seek for the soli-tary wave solutions, such as inverse scattering method, Hopf-Cole transformation, Miura transformations, Dar-boux transformation and Bäcklund transformation [2-5], but solving nonlinear equations is still an important task. In this paper, with the aid of Mathematica, a traveling

wave solution for a class of fifth-order KdV equation

2 5 10 3 20 2 30

t x x x x

u u  uu  u u  u ux (4)

In order to obtain the soliton solution of (4), the soli-tary wave ansatz is assumed as

 

, sechn ,



,

u x t  A B d x ct



(5)

where A is the soliton amplitude, is the width of

the soliton, is the soliton velocity and

d

c B is constant

to be determined later, the unknown index n will be de-

termined during the course of derivation of the solution of Equation (4). From Equation (5), I obtain Equation (6).

With the aid of Mathematica or Maple, from (5) and (6

uating the exponents ), we can get Equation (7).

Now, from Equation (7) eq

1 3 n and 3 2 n leads 1 3 n 3 2n, which gives

2.

n From (7) setting the coeffici

sinh

ents of

3

sech  , _sech5_{sinh and sech}7_{sinh to}

zero, I get Equations (8)-(10).

 





























 













2

2 1

5 3 1 2 1

5

2 1 1

2

2 3 3 3

2 3

, sech , sech sinh .

sech sinh 4 1 sech sinh

4 1 sech sinh 6 1 2 sech sinh

1 2 sech sinh 2 1 2 sech sinh

3 1 1 2 sech

n n

t

n n

x

n n

n n

n

u x t B A u Acdn

u d An An n

An n An n n

An n n A n n n

An n An n n

  

   

   

3

  



 

 

 



  

   

    

     

    



















 

















3

3 3

5 5

2 1 1

3

3 3

3 1

2

2 2

1

sinh

4 1 2 3 sech sinh

1 2 3 4 sech sinh

sech sinh 1 sech sinh

1 2 sech sinh

sech sinh sech sinh

1 sech sinh

sech si

n n

n n

x

n

n n

x x

n n

x

An n n n

An n n n n

u An An n

An n n

u u Ad n An

An n

u Adn

 

 

 

   

 

   

 



 

 

 

 

   

    

  

  

  

 

  nh .



(6)













2

5 3 2

2 3 3 5 5 1

3 5 3 2 5 2 3 3

5 3 5 4 5 5 3

5 5 2 5 3 5 4 5 5 5

10 20 30

30 10 sech sinh

20 8 30 20 10

20 10 2 sech sinh

24 50 35 10 sech sinh

60

t x x x x x

n

n

n

u u uu u u u u

AB dn Acdn ABd n Ad n

ABd n Ad n ABd n Ad n ABd n

Ad n Ad n ABd n

Ad n Ad n Ad n Ad n Ad n

 

 

 







   

   

     

  

    











2 2 3 3 1 2

2 3 2 3 2 2 3 3 3 2

3 1 3

30 sech sinh

20 50 30 sech sinh

3020 sech sinh 0

n

n n

A B d n A d n

A d n A d n A d n

A dn

 

 

 



 



   

 

(7)

(8)

0, (9)

0. (10) Solving the above system by the aid of Wu elimination method [25], I obtain the two solutions

4 ₍₁₁₎

and

2 2 4

30B  c 40Bd 16d 0,

2 2 4

2Bd AB 2Ad 4d

    

2 ₈ 2 ₁₂ 4 A  Ad  d 

2 2

6 , 2 , 56 ,

A d B  d c  d

2 4 2

2 , , 16 .

3

4

A d B  d c  d (12)

Then the soliton solutions of the fifth order KdV equa-is given by

tion

 

2



4



1 , 6 sech 56 2 ,

n

u x t  d d x d t  d2

and

(13)

 

2



4



4 2

2 , 2 sech 16 .

3

u x t  d nd x d t  d (14)

3. Systematic Construction Method

nfinitely Many Conservation L

ifth-Order KdV Equation

We recall the definition [16,23] of a differe (DE

of

aws for

I

F

ntial equation ) that describes a pss. Let 2

M be a two dimensional

differentiable manifold with coordinates



x t .,



A DE for a real function a nec-essary and sufficient conditi for the exi

unctions

(15) depending on u and its derivatives such that the

one-forms

,

 

,

u x t describes

on

pss if it is a stence of dif-ferentiable f

, 1 3,1 2,

ij

f  i  j

1 f11dx f12d ,t 2 f21dx f22d ,t 3 f31dx f32dt

        

(16) isfy the structure equations of a pss, i.e.

sat ,

1 3 2 2 1 3 1 2

d   , d   3, d   . (17)

As a consequence, each solution of the D ovides a local metric on

E pr

2

M , who

DE for is the integrability condition for the problem [14,26]:

se Gaussian curvature is con-stant, equal to −1. Moreover, the above definition is equivalent to saying that u

1

d Ω , ,

2



  

  

 _{  } (18)

 

where d denotes exterior differentiation,   is a col-um r and the 2 × 2 matrix

 

n vecto Ω Ω



_ij, ,i j1, 2



is traceless

2 1 3

1 3 2

 

Take

1

Ω .

2   

  _{ } 

,  (19)

from Equations (18) and (19), we obtain

  

 

d d d d

Ω d d

d d d d

x A t q x B t

S x T t

r x C t x A t





 

 

_{ }

  

 

, ,

x S t T

      (20) where S and T are two 2 × 2 null-trace matrices

,

q S

r

 

 

  _ 

  (21)

.

A B

T

C A

 

  _ 

  (22)

Here  is a parameter, independent of x and t,

while q and r are functions of x and t. Now





2

0 d dΩ Ω d dΩ Ω Ω ,

which requires the vanishing of the two form

ΘdΩ Ω Ω  0, (23) or in component form

0, , 0,

2 2

x

A qC rB

q Aq B B

– 2 2 0

t x

t x

r C Ar C

   

   

  

(

or

24)

11, 12, 31 22 21 32,

21, 22, 11 32 12 31,

31, 32, 11 22 12 21, t x

t x

t x

f f f f f f

f f f f f f

f f f f f f

   

   

   

(25)

where















22 12 32 12 32

1 1 1

, ,



11 31 11 31

2 2

.

2 2 2

1 1

,

q f f r f f

A f B f  f c f  f

   

(26)

Chern and Tenenblat [27] obtain

rectly from the structure equations (17). By suitably choosing and in (24), we shall obtain vari-ous fifth V uation wh

Konno and Wadati introduced the function [28]

ed Equation (24)

di-, di-,

r A B

order Kd

C

eq ich q must satisfy.

1

Γ ,

2

 

 (27)

this function first appeared used and explained in the uations in [11,13], and see also the classical pap

27]. Then Equati :

geometric context of pseudo spherical eq

ers by Sasaki [29] on (20) is reduced and Chern-Tenenblat [

to the Riccati equations

2 _,

r q

x 

_{    }

 (28)

2

2A C B.

t



    

 (29)



 



Гx Гt 2 Г.

c r  cqrB  c Ar (30)

to both sides and using the expression



rГt



Adding

x c

A q rfrom (24), Equation (30) takes the form

 

rГ



A cГ



0,

t x

 

  

  (31)

let us show how an infinite number of conservatio result from these results. The Riccati equations for in the

n laws

Г

-x variable can be rearranged to take the form

 

 

2 Г

Г Г r

r rq r r

x r

      _ _

   (32)

A similar pair of equations can be obtained for the t

derivatives. Expand rГ into a power series in the

verse of

in- so that





1 Г , , _{n n} ,

r x t 

 

x t n ₍₃₃₎ 







 

the n are unknown at this point, however a recursion

relation can be obtained for the n by using (32),

sub-stituting (33) into the Г equation in (32), I find that

 

 



 



1 1 , n n

n x t 

   



2 1 1 , , n n n n

rq x t

x t r r                      





a j n n

 ₍₃₄₎

n x    

Appling the Cauchy product formul

2

 







1

1 , 2

n n

n

n x t n j



n j             



 

in (34), then I obtain

 

 





 

 

1 1 2 , n n

n x t

     



 1 2 1 2 , n n

j n j n j

rq _  _ x t 

     

 

  1 1 2 , , . n n n x x

x t x t

r r

r  r 

                       



   (35)

Now equate powers of  on both sides of this ex-pression to produce the set of recursions,

 

 

1 2

1

1 , , . 2

x n n

n k n k

x

r x t x t n

r         _{ }       (36) 1 , k rq rq       



Substituting (33) into (31), the followin

conservation laws appears g system of

 

 

1 1

, ,

.

n n



n n

This procedure generates an infinite number of con-servation laws for the equation under examination. To obtain conservation laws using (37) in a particular exam-ple using this procedure, let us consider the fifth-order

KdV Equation (4), for Equation (4)

n n

x t x t

A c

t  x r 

            _  _  _{  }



(37) 3 2 5

2 2 3

4 2 2 2

2 2

4x 3x 8 2x 2x 6

c u u uu u

3 3 2 2 4

1, , 6

3 ,

2

2 6 2 ,

6 6 2 .

x x

x

x

x

x x

q r u A u u uu

u u u

B u u u

u

uu u u u u

  _x

                               (38)

nto (24), I obtain the fifth-order tting (38) into (37), it is found that

     

Substituting (38) i KdV Equation (4). Pu

 

 

1 1

, ,

n n n n

n n

x t x t

A c

t  x r 

            _  _  _{  }





Since u

4.

ear wave that possesses remarkable stability properties. Typically, prob- lems that admit soliton solutions are in the form o lution equations that describe how some variable or set of variables evolve in time from a given state. The equa- tio

tions, partial difference equa- tions, and integro-differential equations, as well as cou- pled ODEs of finite order.

In this paper, we considered the construction of exact so

hysics and applied mathematics. Solitons are found in various areas of physics from hydrodynamics and plasma physics, non- linear optics and solid state physics, to field theory and gravitation. NLEEs which describe soliton ph

have a universal character.

A travelling wave of permanent form has already been met; this is the solitary wave solution of the NLEE itself. Such a wave is a special solution of the governing equa-tio

The Soliton equations play a central role in the field of integrable systems and also play a fundamental role in several other areas of mathematics and physics.

REFERENCES

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1 u, 2 0, 3 u , 4 u x,

        

Conclusions

A soliton is a localized pulse-like nonlin

f evo-

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enomena

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