Journal of Applied Mathematics
Volume 2012 (2012), Article ID 896748, 21 pages
http://dx.doi.org/10.1155/2012/896748
Research Article

A New Extended Jacobi Elliptic Function Expansion Method and Its Application to the Generalized Shallow Water Wave Equation

Yafeng Xiao,1 Haili Xue,2 and Hongqing Zhang3

1Department of Mathematics, North University of China, Taiyuan 030051, China
2Software School, North University of China, Taiyuan 030051, China
3School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received 25 October 2012; Accepted 19 November 2012

Academic Editor: Renat Zhdanov

Copyright © 2012 Yafeng Xiao 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 symbolic computation, a new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi elliptic functions, are very effective to uniformly construct more new exact periodic solutions in terms of Jacobi elliptic function solutions of nonlinear partial differential equations. As an application of the method, we choose the generalized shallow water wave (GSWW) equation to illustrate the method. As a result, we can successfully obtain more new solutions. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.