Approximate Homotopy Symmetry Reduction Method: Infinite Series Reductions to Kawahara Equation

The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.     

Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method

<正>This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method.Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders,showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method.The homotopy series solutions to the generalized Kawahara equation are consequently derived.     

Approximate Homotopy Symmetry Reduction Method： Infinite Series Reductions to Kawahara Equation

The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.     

Further Riccati Differential Equations with Elliptic Coefficients and Meromorphic Solutions

We exhibit some families of Riccati differential equations in the complex domain having elliptic coefficients and study the problem of understanding the cases where there are no multivalued solutions. We give criteria ensuring that all the solutions to these equations are meromorphic functions defined in the whole complex plane, and highlight some cases where all solutions are, furthermore, doubly periodic.     

几种辅助方程与非线性发展方程的无穷序列精确解

本文为了获得非线性发展方程新的无穷序列精确解,给出了几种辅助方程的Böcklund变换和解的非线性叠加公式,并构造了一些非线性发展方程新的无穷序列精确解,其中包括无穷序列Jacobi椭圆函数解、无穷序列双曲函数解和无穷序列三角函数解.该方法在构造非线性发展方程无穷序列精确解方面具有普遍意义. 关键词： 辅助方程法 解的非线性叠加公式 无穷序列解 非线性发展方程     

Applications of elliptic functions to ion-acoustic plasma waves

New several classes of exact solutions are obtained in terms of the Weierstrass elliptic function for some nonlinear partial differential equations modeling ion-acoustic waves as well as dusty plasmas in laboratory and space sciences. The Weierstrass elliptic function solutions of the Schamel equation, a fifth order dispersive wave equation and the Kawahara equation are constructed. Moreover, Jacobi elliptic function solutions and solitary wave solutions of the Schamel equation are also given. The stability of some periodic wave solutions is computationally studied.     

Relationship Between Soliton-like Solutions and Soliton Solutions to a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m→1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.     

New Multiple Soliton-like Solutions to (3+1)-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1 )-dimensional Burgers equation with variable coefficients.     

用Riccati方程构造非线性差分微分方程新的精确解

把Riccati方程应用到非线性差分微分方程求解领域,并相结合与一种函数变换,借助符号计算系统Mathematica构造了修正的Volterra方程和一般格子方程新的精确孤立波解和三角函数解. 关键词： Riccati方程 函数变换 非线性差分微分方程 孤立波解     

Peaked Traveling Wave Solutions to a Generalized Novikov Equation with Cubic and Quadratic Nonlinearities

The Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation are the three typical integrable evolution equations admitting peaked solitons. In this paper, a generalized Novikov equation with cubic and quadratic nonlinearities is studied, which is regarded as a generalization of these three well-known studied equations. It is shown that this equation admits single peaked traveling wave solutions, periodic peaked traveling wave solutions, and multi-peaked traveling wave solutions.     

Linear superposition solutions to nonlinear wave equations

The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.     

New Multiple Soliton-like Solutions to （3＋1）-Dimensional Burgers Equation with Variable Coefficients

A new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation, which has more new solutions. More new multiple soliton-like solutions are obtained for the (3 1)-dimensional Burgers equation with variable coefficients.     

A new family of evolution water-wave equations possessing two-soliton solutions

We find a new family of fifth-order water-wave equations having common invariant manifold of the fourth order. These evolution equations are nonintegrable except for two cases corresponding to the Sawada–Kotera and Kaup–Kupershmidt equations. The invariant manifold of the family is an autonomous equation F-VI from the Cosgrove's classification of fourth-order ODEs having the Painlevé property. Two-parameter solutions of the equation F-VI allow to find two-soliton solutions for this family of evolution equations.     

非线性演化方程孤立波的同伦分析法求解

利用同伦分析法求解了Burgers方程，得到了其扭结形孤立波的近似解析解，该解非常接近于相应的精确解.结果表明，同伦分析法可用来求解非线性演化方程的孤立波解.同时，也对所用方法进行了一定扩展，得到了Kadomtsev-Petviashvili(KP)方程的钟形孤立子解.经过扩展后的方法能够更方便地用于求解更多非线性演化方程的高精度近似解析解. 关键词： Burgers方程 同伦分析法 KP方程 孤立波解     

一般变换下的Jacobi椭圆函数展开法及应用

将在行波变换下的Jacobi椭圆函数展开法推广到范围非常广泛的一般函数变换下进行，利用这一方法求得了一些非线性发展方程的精确周期解，这些解包括了在行波变换下所求得的周期解. 证明了一些非线性发展方程的周期解一定是行波解. 关键词： 非线性发展方程 周期解 行波解 Jacobi椭圆函数     

Stair and Step Soliton Solutions of the Integrable (2+1) and (3+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equations

The multiple exp-function method is a new approach to obtain multiple wave solutions of nonlinear partial differential equations (NLPDEs). By this method one can obtain multi-soliton solutions of NLPDEs. In this paper, using computer algebra systems, we apply the multiple exp-function method to construct the exact multiple wave solutions of a (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Also, we extend the equation to a (3+1)-dimensional case and obtain some exact solutions for the new equation by applying the multiple exp-function method. By these applications, we obtain single-wave, double-wave and multi-wave solutions for these equations.     

非线性波方程尖峰孤子解的一种简便求法及其应用

根据尖峰孤子解的特点,提出了一种待定系数法求非线性波方程尖峰孤子解的思路和方法,并利用该方法求解了5个非线性波方程,即CH（Camassa-Holm)方程、五阶KdV-like 方程、广义Ostrovsky方程、组合KdV-mKdV方程和Klein-Gordon方程,比较简便地得到了这些方程的尖峰孤子解.文献中关于CH方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.简要说明了非线性波方程存在尖峰孤子解所须满足的特定条件.该方法也适用于求其他非线性波方程的尖峰孤子解. 关键词： 非线性波方程 尖峰孤子解 待定系数法     

A New Algebraic Method and Its Application to Nonlinear Klein-Gordon Equation

In terms of the solutions of the generalized Riccati equation, a new algebraic method, which contains the terms of radical expression of functions f(ξ), is constructed to explore the new exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to nonlinear Klein-Gordon equation, and some new exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.     

求解非线性差分方程孤立波解的直接代数法

推广了求解非线性差分方程孤立波解的直接代数法.用此方法研究了Hybrid晶格方程，借助于符号计算Maple,得到它的新孤波解.这种方法也可用于求解其他的差分方程. 关键词： 微分-差分方程 Hybrid晶格方程 行波解 孤     

非线性波方程求解的新方法

从Legendre椭圆积分和Jacobi椭圆函数的定义出发，得到了新的变换，并把它用于非线性演化方程的求解.用三个具体的例子，如非线性Klein-Gordon方程、Boussinesq方程和耦合的mKdV方程组，说明了具体的求解步骤.比较方便地得到非线性演化方程或方程组的新解析解，如周期解、孤子解等. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤子解