Classifying Traveling Wave Solutions to the Zhiber-Shabat Equation

doi:10.4236/jamp.2013.12001

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Journal of Applied Mathematics and Physics, 2013, 1, 1-3

http://dx.doi.org/10.4236/jamp.2013.12001 Published Online May 2013 (http://www.scirp.org/journal/jamp)

Classifying Traveling Wave Solutions to the

Zhiber-Shabat Equation

Chunyan Wang*, Xinghua Du

Department of Mathematics, Northeast Petroleum University, Daqing, China

Email: *[email protected]

Received June 6, 2013; revised July 1, 2013; accepted July 10, 2013

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

By the complete discriminatio n system for polynomials, we classify exact traveling wave solutions to the Zhiber-Shabat

equation, and compute some new traveling wave solutions.

Keywords: Traveling Wave Solution; Complete Discrimination System for Polynomials; The Zhiber-Shabat Equation

1. Introduction

The study of exact solutions to nonlinear partial differen-

tial equations is an important component of integrable

systems [1]. Many methods, such as the transformed ra-

tional function method [2], the multiple exp-function

algorithm [3] and the factorization method [4], have been

proposed to find exact traveling wave solutions to nonli-

near partial differential equations. At the same time, Ma

has obtained co mplexiton so lutio ns, a kind of multi-wave

solutions, to some non linear partial differential equations

[5,6]. Liu [7] introduced a simple and efficient method to

give the classification of exact traveling wave solutions

to some nonlinear equations [8].

In this paper, we focus on the Zhiber-Shabat equation

to classify its traveling wave solutions. A. M. Wazwaz [9]

and A. G. Davodi et al. [10] have got some traveling

wave solutions to the Zhiber-Shabat equation. By Liu’s

method, we’ll classify exact traveling wave solutions to

the Zhiber-Shabat equation, and compute some new tra-

veling wave solution s to the equation.

2. Exact Traveling Wave Solutions

The Zhiber-Shabat equation reads as:

uu u

upeqere



  (1)

where 0p, qr are three constants. Take the travel-

ing wave transform a ti on



uukx t





 (2)

the corresponding reduced ordinray differential equation

is given by

uu u

kupe qere



 



  (3)

Furthermore, we take

 where z is a function of

u, and so, we have dduzzu





 Substituting these

terms into Equation (2) yields

uu u

k zpeqere







  (4)

Using the method of the variation of constants, the

general solution of Equation (3) is given by

222

uuu

qr pc

zue ee

kk kk









  (5)

where c is an arbitrary constant. Thus th e general solu-

tion of Equati o n (4 ) is



022

uuu

kk kk

ee e

 







 



 (6)

We take the transformation u

ve, the corresponding

integral becomes



022

kkkk

vvv









 



 (7)

Denote





210

wwdwdwd



 (8)

where

13 23

pcp

wvd

kkk

qp r

kk k



 







 



 

 











(9)

*Corresponding autho

C. Y. WANG, X. H. DU

The complete discrimination system for





w is

given by



2012

27 4

27 3

dddd



 





 

(10)

Case 1. 1

00D :

We have



()Fw ww







 . When





, we have



01ln w







 

 

 

 (11)



02arctan w











 



 (12)

The corresponding solutions are



[tanh ]























 





(13)



coth 2



 





























(14)



sec 2



 



















 









(15)

Case 2. 1

00D :

Then

 

Fw w



 . The solution is given by



ln p





































(16)

Case 3. 1

00D :

Then

 

Fw www





  with





. Therefore, we have





0dw

ww w







 



 (17)

When w





 , we obtain a new traveling wave

solution



13 13

sn 2

ukk







 

 



 



 

 













 













(18)

When w



, we obtain another new traveling wave

solution



sn 0

cn 0































































(19)

where











.

Case 4. 0



:

Then













40Fwwwpw qpq







. The

corresponding integral bec om es



wwpwq

 



 



 (20)

When w



, we obtain the following new traveling

wave solution









1cn

pq k







 





























 











(21)

where

212



















3. Acknowledgements

The project is supported by the opening fund of Key

Laboratory of Enhan ced Oil and Gas Recover y of Edu ca-

tion Ministry.

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