应用新展式法求非线性发展方程的精确解

The exp(−ϕ(ξ))-expansion Method applied to Nonlinear Evolution Equations

Mei-mei Zhao∗†,Chao-Li

School of Mathematics and Statistics,Lanzhou University

Lanzhou,Gansu730000,P.R.of China

Abstract

By using exp(−ϕ(ξ))-expansion method,we have obtained more travelling wave solu-tions to the mKdV equation,the Drinefel’d-Sokolov-Wilson equations,the Variant Boussinesq equations and the Coupled Schr¨o dinger-KdV system.The proposed method also can be used for many other nonlinear evolution equations.

Keywords exp(−ϕ(ξ))-expansion method,Homogeneous balance,Travelling wave solu-tions,Solitary wave solutions,MKdV equation,Drinefel’d-Sokolov-Wilson equations,Variant Boussinesq equations,Coupled Schr¨o dinger-KdV system.

1Introduction

It is well known that nonlinear evolution equations are involved in manyfields from physics to biology,chemistry,mechanics,etc.As mathematical models of the phenomena,the inves-tigation of exact solutions to nonlinear evolution equations reveals to be very important for the understanding of these physical problems.Understanding this importance,during the past four decades or so,many mathematicians and physicists have being paid special attention to the development of sophisticated methods for constructing exact solutions to nonlinear evo-lution equations.Thus,a number of powerful methods has been presented such as the inverse scattering transform[1],the B¨a cklund and the Darboux transform[2-5],the Hirota[6],the trun-cated painleve expansion[7],the tanh-founction expansion and its various extension[8-10],the Jacobi elliptic function expansion[11,12],the F-expansion[13-16],the sub-ODE method[17-20],the homogeneous balance method[21-23],the sine-cosine method[24,25],the rank anal-ysis method[26],the ansatz method[27-29],the exp-function expansion method[30],Algebro-geometric constructions method[31]and so on.

In the present paper,we shall proposed a new method which is called exp(−ϕ(ξ))-expansion method to seek travelling wave solutions of nonliear evolution equations.the ∗Corresponding Author.

†E-mail address:yunyun1886358@(M.Zhao).

1

main idears of the proposed method are that the travelling wave solutions of a nonliear evo-lution equation can be expressed by a polynomial in exp(−ϕ(ξ)),whereϕ(ξ)satisfies ODE (see Eq.(5)in section2),ξ=x−V t,the degree of the polynomial can be determined by con-sidering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in a given nonliear evolution equation,and the coefficients of the polynomial from the process of using the proposed method.It will be seen that more travelling wave solutions of many nonlinear evolution equations can be obtained by using the exp(−ϕ(ξ))-expansion method.

The paper is organized as follows:In section2,we will recall the exp(−ϕ(ξ))-expansion method.In section3,we will illustrate the method in detail with the mKdV equation, the Drinefel’d-Sokolov-Wilson equations,the variant Boussinesq equations and the Coupled Schr¨o dinger-KdV system.In section7,the features of the exp(−ϕ(ξ))-expansion method will be briefly summarized.

2Description of the exp(−ϕ(ξ))-expansion method

In the following,we will outline the main steps of exp(−ϕ(ξ))-expansion method.

Consider a nonlinear equation,say in two independent variable x and t,is given by

P(u,u t,u x,u tt,u xt,u xx,...)=0,(1)

where u=u(x,t)is an unknown function,P is a polynomial in u=u(x,t)and its various partial derivatives,in which the highest order derivatives and nonlinear terms are involved. Step1Combining the independent variable x and t into one variableξ=x−V t,we suppose that

u(x,t)=u(ξ),ξ=x−V t,(2)

the travelling wave variable(2)permits us reducing Eq.(1)to an ODE for u=u(ξ)

P(u,−V u ,u ,V2u ,−V u ,u ,...),(3)

Step2Suppose that the solution of ODE(3)can be expressed by a polynomial in exp(−ϕ(ξ)) as follows

u(ξ)=αm(exp(−ϕ(ξ)))m+...,(4)

whereϕ(ξ)satisfies the ODE in the form

ϕ (ξ)=exp(−ϕ(ξ))+µexp(ϕ(ξ))+λ,(5)

the solutions of ODE(5)are

Whenλ2−4µ>0,µ=0,

ϕ(ξ)=ln(−

2tanh(

√

λ2−4µ

2

(ξ+C1))−λ

2µ

),(6) 2