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
Received June 6, 2013; revised July 1, 2013; accepted July 10, 2013
Copyright © 2013 Chunyan Wang, Xinghua Du. 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
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:
20 uu u
xt
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
2
0
uu u
kupe qere 
 

  (3)
Furthermore, we take
uz
 where z is a function of
u, and so, we have dduzzu


 Substituting these
terms into Equation (2) yields
2
d0
d
uu u
z
k zpeqere
u



  (4)
Using the method of the variation of constants, the
general solution of Equation (3) is given by
2
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
22
d
qp uuu
c r
kk kk
u
ee e
 



 

 (6)
We take the transformation u ve, the corresponding
integral becomes

022
32
d
pq
cr
kkkk
v
vvv




 

 (7)
Denote


32
210
F
wwdwdwd

 (8)
where
13 23
2
13
10
22
,
22
pcp
wvd
kkk
qp r
dd
kk k

 




 

 
 





(9)
*Corresponding autho
r
.

C. Y. WANG, X. H. DU 2
The complete discrimination system for


F
w is
given by

2
33
2 13 2012
2
2
11
2
27 4
27 3
3
dd
dddd
d
Dd

 


 
(10)
Case 1. 1
00D :
We have

2
()Fw ww



 . When

 , we have

01ln w
w



 
 
 
 (11)

02arctan w






 

 (12)
The corresponding solutions are
 
13
1
20
2
ln
1
[tanh ]
2
p
uk











 


(13)
 
13
2
20
2
ln
1
coth 2
p
uk

 














(14)


13
3
20
2
ln
1
sec 2
p
uk

 









 




(15)
Case 2. 1 00D :
Then
 
3
Fw w

 . The solution is given by

13
2
0
24
ln p uk 





 
 


  



(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
20
22
ln
sn 2
pp
ukk
k



 
 

 

 
 






 






(18)
When w

, we obtain another new traveling wave
solution


13
2
ln
sn 0
2
cn 0
2
p
uk
k
k
















 


















(19)
where
2
k





.
Case 4. 0

:
Then






22
40Fwwwpw qpq 


. The
corresponding integral bec om es



02
dw
wwpwq
 

 

 (20)
When w

, we obtain the following new traveling
wave solution



 
13
2
14
20
2
2
ln
2
1cn
p
uk
pq
pq k
pq



 














 





(21)
where
212
1
22
p
k
pq










.
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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