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

Yafeng Xiao,Hongqing Zhang,Haili Xue

doi:10.1155/2012/896748

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.

Highlights

In recent years, the nonlinear partial differential equations NPDEs are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, and so forth
With the development of soliton theory, There has been a great amount of activities aiming to find methods for exact solutions of nonlinear differential equations, such as Backlund transformation, Darboux transformation, Cole-Hopf transformation, similarity reduction method, variable separation approach, Exp-function method, homogeneous balance method, varied tanh methods, and varied Jacobi elliptic function methods 1– 21
The direct ansatz method 8–21 provides a straightforward and effective algorithm to obtain such particular solutions for a large number of nonlinear partial differential equations, in which the starting point is the ansatz that the solution sought is Journal of Applied Mathematics expressible as a finite series of special function, such as tanh function, sech function, tan function, sec function, sine-cosine function, Weierstrass elliptic function, theta function, and Jacobi elliptic function

Summary

Introduction

The nonlinear partial differential equations NPDEs are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, and so forth. The direct ansatz method 8–21 provides a straightforward and effective algorithm to obtain such particular solutions for a large number of nonlinear partial differential equations, in which the starting point is the ansatz that the solution sought is Journal of Applied Mathematics expressible as a finite series of special function, such as tanh function, sech function, tan function, sec function, sine-cosine function, Weierstrass elliptic function, theta function, and Jacobi elliptic function. A new Jacobi elliptic function expansion method is presented by means of a new general ansatz and is more powerful to uniformly construct more new exact doubly-periodic solutions in terms of Jacobi elliptic functions of nonlinear partial differential equations.

Summary of the Extended Jacobi Elliptic Function Expansion Method

Exact Solutions of the GSWW Equation

Conclusion and More General