The(G′/G)-Expansion Method and Its Application for Higher-Order Equations of KdV (III)

Huizhang Yang,Wei Li,Biyu Yang

doi:10.1155/2014/384969

Abstract

New exact traveling wave solutions of a higher-order KdV equation type are studied by the(G′/G)-expansion method, whereG=G(ξ)satisfies a second-order linear differential equation. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. The property of this method is that it is quite simple and understandable.

Highlights

Nonlinear evolution equations (NLEEs) are widely used as models to describe the complex physical phenomena and a troublesome and tedious but very important problem is to find exact solutions of NLEEs
Bekir [14] was concerned with this method to study nonlinear evolution equations for constructing wave solutions
The further developed methods named the generalized (G󸀠/G)expansion method [17], the modified (G󸀠/G)-expansion method [18], the extended (G󸀠/G)-expansion method [19], the improved (G󸀠/G)-expansion method [20], the generalized and improved (G󸀠/G)-expansion method [21], and the (G󸀠/G, 1/G)-expansion method [22] have been proposed for constructing exact solutions to NLEEs

Summary

Introduction

Nonlinear evolution equations (NLEEs) are widely used as models to describe the complex physical phenomena and a troublesome and tedious but very important problem is to find exact solutions of NLEEs. In recent years, more and more researchers investigated exact traveling wave solutions of NLEEs and lots of effective methods have been proposed, such as the inverse scattering method [1], the Backlund transform method [2, 3], the Darboux transform method [4], the Hirota bilinear transformation method [5], the Expfunction method [6, 7], the tanh-function method [8, 9], the Weierstrass elliptic function method [10], and the Jacobi elliptic function expansion method [11]. The (G󸀠/G)-expansion method, firstly presented by Wang et al [12], has been widely used to search for various kinds of exact solutions of NLEEs. For instance, Malik et al [13] applied the (G󸀠/G)-expansion method in getting traveling wave solutions of some nonlinear partial differential equations. The further developed methods named the generalized (G󸀠/G)expansion method [17], the modified (G󸀠/G)-expansion method [18], the extended (G󸀠/G)-expansion method [19], the improved (G󸀠/G)-expansion method [20], the generalized and improved (G󸀠/G)-expansion method [21], and the (G󸀠/G, 1/G)-expansion method [22] have been proposed for constructing exact solutions to NLEEs

Objectives

Conclusion