非线性演化方程的孤立波解

用齐次平衡原则和辅助微分方程方法得到了6个重要的n次非线性演化方程的孤立波解.辅助微分方程方法的主要思想是借助简单的可解微分方程的解去构造复杂的非线性演化方程的行进波解.这里简单的可解微分方程称为辅助微分方程.本文使用的辅助方程有双曲正割幂型解或双曲正切幂型解.     

On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation are successfully obtained, which contain soliton-like and periodic-like solutions This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

New periodic wave solutions via extended mapping method

In this paper, an extended mapping method with a computerized symbolic computation is used for constructing new periodic wave solutions for two nonlinear evolution equations arising in mathematical physics, namely, generalized nonlinear Schroedinger equation and generalized-Zakharov equations. As a result, many exact travelling wave solutions are obtained which include new periodic wave solutions, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also applied to other nonlinear evolution equations.     

New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation

The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations.     

New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation

The repeated homogeneous balance method is used to construct new exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain soliton-like and periodic-like solutions. This method is straightforward and concise, and it can be also applied to other nonlinear evolution equations.     

New periodic wave solutions for nonlinear evolution equations with variable coefficients via mapping method

An extended mapping method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for nonlinear evolution equations arising in physics, namely, generalized Zakharov Kuznetsov equation with variable coefficients. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations with variable coefficients arising in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Construction of periodic and solitary wave solutions by the extended Jacobi elliptic function expansion method

In this paper, an extended Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the exact periodic solutions of some polynomials or nonlinear evolution equations. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in nonlinear mathematical physics. As a result, many exact travelling wave solutions are obtained which include new solitary or shock wave solution and envelope solitary and shock wave solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

一类非线性演化方程新的显式行波解

借助Mathematica软件和吴方法,采用双曲函数法,获得了一类非线性演化方程u_tt+au_xx+bu+cu2+du3=0的多组行波解,其中包括周期解与孤子解.这种方法也适用于其他非线性方程或方程组.     

Travelling wave solutions of nonlinear evolution equations by using the first integral method

In this paper, we established travelling wave solutions for some (2 + 1)-dimensional nonlinear evolution equations. The first integral method was used to construct travelling wave solutions of nonlinear evolution equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The first integral method presents a wider applicability for handling nonlinear wave equations.     

累次齐次平衡法及其应用

在求非线性偏微分方程精确解的过程中两次使用了齐次平衡法(称为累次齐次平衡法),解决了齐次平衡法求解少的不足,从而改进了齐次平衡法.以高阶(2+1)维Kadomtsev-Petviashvili方程和变异的Boussinesq方程为应用实例,说明使用累次齐次平衡法可以求得大量的精确解,其中许多解是新解或覆盖了其他方法所得的解.方法可应用于大量的非线性物理模型.     

The homogeneous balance method and its application to the Benjamin-Bona-Mahoney (BBM) equation

We make use of the homogeneous balance method and symbolic computation to construct new exact traveling wave solutions for the Benjamin-Bona-Mahoney (BBM) equation. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

非线性系统的精确解

Based on the computerized symbolic, a new generalized tanh functions method is used for constructing exact travelling wave solutions of nonlinear partial differential equations （PDES） in a unified way. The main idea of our method is to take full advantage of an auxiliary ordinary differential equation which has more new solutions. At the same time, we present a more general transformation, which is a generalized method for finding more types of travelling wave solutions of nonlinear evolution equations （NLEEs）. More new exact travelling wave solutions to two nonlinear systems are explicitly obtained.     

齐次平衡法若干新的应用

齐次平衡法是求非线性发展方程孤波解的一种有效方法．该文将以KdV方程为例把齐次平衡法向三个方面拓广应用：1）获得非线性发展方程新的具有更为丰富形式的精确解;2）寻找非线性发展方程的Backlund变换、Lax表示;3）求非线性发展方程的对称性约化和相似解．     

New exact travelling wave solutions for some nonlinear evolution equations, Part II

By using new solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equation. By this method some nonlinear evolution equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.     

二维色散长波方程组的精确解

利用齐次平衡法给出了二维色散长波方程组的定态解、孤立波解与非孤立波解等几种显式精确解。这个方法也可用来寻找其它非线性发展方程的不同类型的精确解。     

双函数法及一类非线性发展方程的精确行波解

给出一种求解非线性发展方程精确行波解的新方法：双函数法。使用此方法，获得了一类非线性发展方程的许多精确行波解，其中包括孤波解和周期解，推广了文献用其它方法取得的结果，同时还获得了许多新的弧波解和周期解，借助于Mathemat-ica，此方法能部分地在计算机上实现。     

非线性发展方程行波解的符号计算

Abstract. A Riccati equation involving a parameter and symbolic computation are used to uni-formly construct the different forms of travelling wave solutions for nonlinear evolution equa-tions. It is shown that the sign of the parameter can be applied in judging the existence of vari-ous forms of travelling wave solutions. An efficiency of this method is demonstrated on some e-quations,which include Burgers-Huxley equation,Caudrey-Dodd-Gibbon-Kawada equation,gen-eralized Benjamin-Bona-Mahony equation and generalized Fisher equation.     

Exact solutions of nonlinear Schrödinger equation by using symbolic computation

The (G′/G,1/G)‐expansion method and (1/G′)‐expansion method are interesting approaches to find new and more general exact solutions to the nonlinear evolution equations. In this paper, these methods are applied to construct new exact travelling wave solutions of nonlinear Schrödinger equation. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. It is shown that the proposed methods provide a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Copyright © 2015 John Wiley & Sons, Ltd.     

New exact travelling wave solutions for the Kawahara and modified Kawahara equations

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact travelling wave solutions for nonlinear evolution equations. By this method the Kawahara and the modified Kawahara equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.     

一类非线性波动方程的显式精确解

本文用直接方法和假设的一种结合求出了一类较广泛的非线性波动方程ｕｔｔ－ａ１ｕｘｘ＋ａ２ｕｔ＋ａ３ｕ＋ａ４ｕＳ＾２＋ａ５ｕ＾３＝０的一些显式精确行波解,贱个有重要的非线性数学物理方程,如φ＾４方程,Ｋｌｅｉｎ－Ｇｏｒｄｏｎ方程,Ｓｉｎｅ－Ｇｏｒｄｏｎ方程,及Ｓｉｎｈ－Ｇｏｒｄｏｎ方程的近似,Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ－Ｈｉｇｇｓ方程,Ｄｕｆｆｉｎｇ方程,非线性电报方程等都可作为该方程的特殊情形得