Solving two fifth order strong nonlinear evolution equations by using the GG′-expansion method

doi:10.1016/j.cnsns.2009.10.006

Communications in Nonlinear Science and Numerical Simulation

Volume 15, Issue 9, September 2010, Pages 2337-2343

https://doi.org/10.1016/j.cnsns.2009.10.006 Get rights and content

Abstract

In this paper, new $(\frac{G}{G^{'}})$ -expansion method is successfully implemented to find travelling wave solutions for two fifth order strong nonlinear evolution equations whose balance is not positive integers. As a result, some new exact solutions with parameters are obtained. Compared with other methods, this method is direct, concise, effective and easy to calculate, and it is a powerful mathematical tool for obtaining exact travelling wave solutions of nonlinear evolution equations and can be used to solve other nonlinear partial differential equations in mathematical physics.

Introduction

The research area of nonlinear evolution equation has been very active for the past few decades. There are various kinds of nonlinear evolution equations that appear in various areas of physical and mathematical sciences. Much effort has been made on the construction of exact solutions of nonlinear equations, for their important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena appears in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equation. In recent years, the powerful and efficient methods to find analytic solutions of nonlinear equations have drawn a lot of interest by a diverse group of scientists. Such as tanh-function method [1], [3], homogeneous balance method [1], [2], extended tanh-function method [1], [10], [13], auxiliary equation method [11], [15], Painleve expansion method [2], [3], [9], Jacobi elliptic function method [1], [2], [3], sine–cosine function method [12], Exp-function method [1], [2], [14] and so on were used to develop nonlinear dispersive and dissipative problems.

Recently, Wang et al. [4] proposed the $(\frac{G^{'}}{G})$ -expansion method and showed that it is powerful for finding analytic solutions of nonlinear evolution equations. Next, Bekir [5] applied the method to some nonlinear evolution equations gaining travelling wave solutions. Later, Zhang et al. [6] have generalized the method to obtain non-travelling wave solutions and coefficient function solutions and Zhang et al. [7] further extended the method to solve an evolution equation with variable coefficients.

In this paper, we improved the method of Wang et al. [4] and give a new generalized form as $(\frac{G}{G^{'}})$ -expansion and successfully obtained some new exact solutions of (1 + 1)-dimensional Kundu equation [8], [15] and (1 + 1)-dimensional Schroˇdinger equation [8], [16], [17], [18], [19]. The results show that this method is not only simple, effective and straightforward, but also can be used for many other nonlinear evolution equations in mathematical physics.

Section snippets

Description of the generalized $(\frac{G}{G^{'}})$ -expansion method

In this section we describe the $(\frac{G}{G^{'}})$ -expansion method for finding travelling wave solutions of nonlinear evolution equations as follow. For a given nonlinear partial differential equation $P (u, u_{t}, u_{x}, u_{y}, u_{z}, \dots, u_{xt}, u_{yt}, u_{zt}, \dots, u_{tt}, u_{xx}, u_{yy}, x_{zz}, \dots) = 0 .$

Our method mainly consist four steps:

Step 1: We seek complex solutions of Eq. (1) as the following form $u (x, y, z, t) = u (ξ), ξ = k_{1} x + k_{2} y + k_{3} z + k_{4} t,$
where $k_{j} (j = 1, 2, 3, 4)$ are real constants. Under the transformation (2), Eq. (1) becomes an ordinary differential equation $Q (U, U$

(1 + 1)-dimensional Kundu equation [8,15]

${iu}_{t} + u_{xx} + β | u |^{2} u + γ | u |^{4} u + i α (| u |^{2} u)_{x} + is (| u |^{2})_{x} u = 0 .$

We may choose the following travelling wave transformation $u (x, t) = e^{i (φ (ξ) - θ t)} v (ξ), ξ = x - ω t,$ where $ω, θ$ are constant determined later. Substituting (7) into (6), we have $- ω v^{'} + 2 v^{'} φ^{'} + v φ^{″} + (3 α + 2 s) v^{2} v^{'} = 0,$ $v (φ^{'} ω + θ) + v^{″} - v (φ^{'})^{2} - α v^{3} φ^{'} + β v^{3} + γ v^{5} = 0 .$

If setting $φ^{'} = \frac{ω}{2} - \frac{3 α + 2 s}{4} v^{2},$ substituting (10) into (8), (9), we have $v^{″} + (\frac{ω^{2}}{4} + θ) v + (β - \frac{α ω}{2}) v^{3} + (γ + \frac{1}{16} (3 α + 2 s) (α - 2 s)) v^{5} = 0 .$ According to Step 2, we get $m = \frac{1}{2}$ . Therefore, we can write the solution of Eq. (11) in the form $v (ξ) = A {(\frac{G}{G^{'}})}^{\frac{1}{2}},$ where A is

Conclusions and discussion

A new form of the $(\frac{G}{G^{'}})$ -expansion method has been successfully implemented to find travelling wave solutions of (1 + 1)-dimensional Kundu equation and (1 + 1)-dimensional Schroˇdinger equation in this paper. As a result, some exact travelling wave solutions with parameters $c_{1}$ and $c_{2}$ had been obtained, where included some new solutions. The result show that it is also a promising method to solve other nonlinear partial differential equations in mathematical physics. More importantly, this method can

Acknowledgments

This work is supported in part by the Doctoral Unit Foundation of Ministry of Education of China (Grant No. 20070128001), the High Education Science Research Program of Inner Mongolia (Grant No. NJZY07066).

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Solving two fifth order strong nonlinear evolution equations by using the GG′-expansion method

Abstract

Introduction

Section snippets

Description of the generalized GG′-expansion method

(1 + 1)-dimensional Kundu equation [8,15]

Conclusions and discussion

Acknowledgments

Phys Lett A

Phys Lett A

Phys Lett A

Phys Lett A

Chaos, Solitons and Fractals

Phys Lett A

Chaos, Solitons and Fractals

Appl Math Comput

Travelling wave solutions of the nonlinear mathematics and physics equation

The method of nonlinear mathematics and physics

Nonlinear equation of the physics

Solving two fifth order strong nonlinear evolution equations by using the $(\frac{G}{G^{'}})$ -expansion method

Description of the generalized $(\frac{G}{G^{'}})$ -expansion method