Elsevier

Chaos, Solitons & Fractals

Volume 30, Issue 3, November 2006, Pages 700-708
Chaos, Solitons & Fractals

Exp-function method for nonlinear wave equations

https://doi.org/10.1016/j.chaos.2006.03.020Get rights and content

Abstract

In this paper, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd–Bullough–Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.

Introduction

Recently many new approaches to nonlinear wave equations have been proposed, for example, tanh-function method [1], [2], [3], [4], [5], [6], F-expansion method [7], [8], [9], Jacobian elliptic function method [10], [11], [12], variational iteration method [13], [14], Adomian method [15], [16], [17], [18], variational approach [19], [20], [21], and homotopy perturbation method [22], [23], [24]. All methods mentioned above have limitation in their applications. In this paper we suggest a novel method called Exp-function method (or Exp-method for short) to search for solitary solutions, compact-like solutions and periodic solutions of various nonlinear wave equations.

Section snippets

Basic idea of Exp-function method

In order to illustrate the basic idea of the suggested method, we consider first the following nonlinear dispersive equation of the form [13], [25], [26], [27]:ut+u2ux+uxxx=0.This equation is called modified KdV equation, which arises in the process of understanding the role of nonlinear dispersion and in the formation of structures like liquid drops, and it exhibits compactons: solitons with compact support.

Introducing a complex variation η defined asη=kx+ωt.We haveωu+ku2u+k3u=0,where prime

An example

Now we consider the Dodd–Bullough–Mikhailov equation [4]uxt+eu+e-2u=0.This equation plays a significant role in many scientific applications such as solid state physics, nonlinear optics and quantum field theory. By the transformation u = ln v, Eq. (26) becomesvvxt-vxvt+v3+1=0.Introducing a complex variation η defined as η = kx + ωt, we havef(v)kωvv-kω(v)2+v3+1=0,where prime denotes the differential with respect to η.

We suppose that the solution of Eq. (28), can be expressed asu(x,t)=acexp[c(kx+ωt)]

Conclusion

We give a very simple and straightforward method called Exp-function method for nonlinear wave equations. The suggest method has some pronounced merits:

  • (1)

    The method leads to both the generalized solitonary solutions and periodic solutions;

  • (2)

    The solution procedure, by help of Matlab, is of utter simplicity, and can be easily extended to all kinds of nonlinear equations.

The Exp-function method might become a promising and powerful new method for nonlinear equations.

Acknowledgement

This work is supported by Program for New Century Excellent Talents in University and the William M.W. Wong Engineering Research Fund.

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