New Exact Jacobi Elliptic Function Solutions for the Coupled Schrödinger-Boussinesq Equations

Baojian Hong,Dianchen Lu

doi:10.1155/2013/170835

Abstract

A general algebraic method based on the generalized Jacobi elliptic functions expansion method, the improved general mapping deformation method, and the extended auxiliary function method with computerized symbolic computation is proposed to construct more new exact solutions for coupled Schrödinger-Boussinesq equations. As a result, several families of new generalized Jacobi elliptic function wave solutions are obtained by using this method, some of them are degenerated to solitary wave solutions and trigonometric function solutions in the limited cases, which shows that the general method is more powerful than plenty of traditional methods and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

Highlights

Nonlinear partial differential equations (NLPDEs) are widely used to describe complex physical phenomena arising in the world around us and various fields of science
The solution procedure of this method, by the help of Matlab or Mathematica, is of utmost simplicity, and this method can be extended to all kinds of NLPDEs
Our method proposed here is more general than the G󸀠/G method [8], the extended Riccati equation rational expansion method [12], the extended auxiliary function method [13], the generalized Jacobi elliptic functions expansion method [16, 17], and many other algebra expansion methods [6, 7, 11, 18,19,20,21]

Summary

Introduction

Nonlinear partial differential equations (NLPDEs) are widely used to describe complex physical phenomena arising in the world around us and various fields of science. With the development of soliton theory, many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as homotopy perturbation method [1], nonperturbative method [2], homogeneous balance method [3], Backlund transformation [4], Darboux transformation [5], extended tanh-function method [6], extended F-expansion method [7], G󸀠/G method [8], exp-function method [9], sine-cosine method [10], Jacobi elliptic function method [11], extended Riccati equation rational expansion method [12], extended auxiliary function method [13], and other methods [14, 15]. In [16, 17], Hong proposed a generalized Jacobi elliptic functions expansion method to obtain generalized exact solutions of NLPDEs. In [18], Hong and Lu proposed an improved general mapping deformation method which is more general than many other algebraic expansion methods [19, 20]. We will propose the general algebraic method which contained the two methods [16,17,18] to obtain several new families of exact solutions for the coupled SchrodingerBoussinesq equations

Summary of the General Algebraic Method

Exact Solutions to the Coupled SchrödingerBoussinesq Equations

Conclusion