 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) Copyright © 2013 SciRes. JAMP Classifying Traveling Wave Solutions to the Zhiber-Shabat Equation Chunyan Wang*, Xinghua Du Department of Mathematics, Northeast Petroleum University, Daqing, China Email: *chunyanmyra@163.com 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 . Many methods, such as the transformed ra- tional function method , the multiple exp-function algorithm  and the factorization method , 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  introduced a simple and efficient method to give the classification of exact traveling wave solutions to some nonlinear equations . In this paper, we focus on the Zhiber-Shabat equation to classify its traveling wave solutions. A. M. Wazwaz  and A. G. Davodi et al.  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: 20uu uxtupeqere  (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 20uu ukupe 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 2d0duu uzk zpeqereu  (4) Using the method of the variation of constants, the general solution of Equation (3) is given by 2222uuuqr pczue eekk kk  (5) where c is an arbitrary constant. Thus th e general solu- tion of Equati o n (4 ) is 02222dqpuuucrkk kkuee e   (6) We take the transformation uve, the corresponding integral becomes 02232dpqcrkkkkvvvv  (7) Denote 32210Fwwdwdwd (8) where 13 232131022,22pcpwvdkkkqp rddkk k     (9) *Corresponding author. C. Y. WANG, X. H. DU Copyright © 2013 SciRes. JAMP 2 The complete discrimination system for Fw is given by 23321320122211227 427 33dddddddDd   (10) Case 1. 100D : We have 2()Fw ww . When , we have 01ln ww    (11) 02arctan w  (12) The corresponding solutions are 131202ln1[tanh ]2puk  (13) 132202ln1coth 2puk  (14) 133202ln1sec 2puk   (15) Case 2. 100D : Then  3Fw w . The solution is given by 132024ln puk (16) Case 3. 100D : Then  Fw www  with . Therefore, we have 0dwww w  (17) When w , we obtain a new traveling wave solution 13 132022lnsn 2ppukkk       (18) When w, we obtain another new traveling wave solution 132lnsn 02cn 02pukkk (19) where 2k. Case 4. 0: Then 2240Fwwwpw qpq. The corresponding integral bec om es 02dwwwpwq   (20) When w, we obtain the following new traveling wave solution 132142022ln21cnpukpqpq kpq   (21) where 212122pkpq. 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. REFERENCES  W. X. Ma, “Integrabil ity,” In: A. 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