A Series of Soliton-like and Double-like Periodic Solutions of a （2＋1）-Dimensional Asymmetric Nizhnik-Novikov-Vesselov Equation

We generalize the algebraic method presented by Fan [J.Phys. A: Math. Gen. 36 (2003) 7009)] to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations (NPDE). As an application of the method, we choose a (2 1)-dimensional asymmetric Nizhnik Novikov Vesselov equation and successfully construct new and more general solutions including a series of nontraveling wave and coefficient functions‘soliton-like solutions, double-like periodic and trigonometric-like function solutions.     

Solutions to the Complex Korteweg-de Vries Equation: Blow-up Solutions and Non-Singular Solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, includ- ing blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Kortcweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the com- plex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.     

A Series of Exact Solutions of （2＋l）-Dimensional CDGKS Equation

An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions. Among them, the Jacobi periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh method, the method used here can give new and more general solutions. More importantly, this method provides a guldeline to classifj, the various types of the solution according to some parameters.     

New explicit exact solutions for a generalized Hirota—Satsuma coupled KdV system and a coupled MKdV equation

In this paper, we make use of a new generalized ansatz in the homogeneous balance method, the well-known Riccati equation and the symbolic computation to study a generalized Hirota--Satsuma coupled KdV system and a coupled MKdV equation, respectively. As a result, numerous explicit exact solutions, comprising new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.     

An Automated Algebraic Method for Finding a Series of Exact Travelling Wave Solutions of Nonlinear Evolution Equations

Based on a type of elliptic equation,a new algebraic method to construct a series of exact solutions for nonlinear evolution equations is proposed,meanwhile,its complete implementation TRWS in Maple is presented.The TRWS can output a series of travelling wave solutions entirely automatically,which include polynomial solutions,exponential function solutions,triangular function solutions,hyperbolic function solutions,rational function solutions,Jacobi elliptic function solutions,and Weierstrass elliptic function solutions.The effectiveness of the package is illustrated by applying it to a variety of equations.Not only are previously known solutions recovered but also new solutions and more general form of solutions are obtained.     

New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota－Satsuma KdV system

In this paper,we study the generalized coupled Hirota-Satsuma KdV system by using the new generalized transformation in homogeneous balance method.As a result,many explicit exact solutions,which contain new solitary wave solutions,periodic wave solutions,and the combined formal solitary wave solutions,and periodic wave solutions ,are obtained.     

Applications of Elliptic Equation to Nonlinear Coupled Systems

The elliptic equation is taken as a transformation and applied to solve nonlinear coupled systems. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions, periodic wave solutions and so on, so this method can be taken as a unified method in solving nonlinear coupled systems.     

Auto-Backlund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients

By using the extended homogeneous balance method, a new auto-Ba^ecklund transformation(BT) to the generalized Kadomtsew-Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.     

Auto-Backlund Transformation and Analytic Solutions of （2＋1）-Dimensional Boussinesq Equation

Using the truncated Painleve expansion, symbolic computation, and direct integration technique, we study analytic solutions of （2＋1）-dimensional Boussinesq equation. An auto-Backlund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass ＆ function. Some of them are novel.     

Solving the AKNS Hierarchy by Its Bilinear Form： Generalized Double Wronskian Solutions

Through the Wronskian technique, a simple and direct proof is presented that the AKNS hierarchy in the bilinear form has generalized double Wronskian solutions. Moreover, by using a unified way, soliton solutions, rational solutions, Matveev solutions and complexitons in double Wronskian form for it are constructed.     

(2+1) 维Broer-Kau-Kupershmidt方程一系列新的精确解

借助于符号计算软件Maple，通过一种构造非线性偏微分方程(组)更一般形式精确解的直接方法即改进的代数方法，求解(2+1) 维 Broer-Kau-Kupershmidt方程，得到该方程的一系列新的精确解，包括多项式解、指数解、有理解、三角函数解、双曲函数解、Jacobi 和 Weierstrass 椭圆函数双周期解. 关键词： 代数方法 (2+1) 维 Broer-Kau-Kupershmidt 方程 精确解 行波解     

A Series of Exact Solutions of (2+1)-Dimensional CDGKS Equation

An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the (2 1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions. Among them, the Jacobi periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh method, the method used here can give new and more general solutions. More importantly, this method provides a guideline to classify the various types of the solution according to some parameters.     

New exact solutions to the high dispersive cubic–quintic nonlinear Schrödinger equation

The complete discrimination system method is employed to find exact solutions for a dispersive cubic–quintic nonlinear Schrödinger equation with third order and fourth order time derivatives. As a result, we derive a range of solutions which include triangular function solutions, kink solitary wave solutions, dark solitary wave solutions, Jacobian elliptic function solutions, rational function solutions and implicit analytical solutions. Numerical simulations are presented to visualize the mechanism of Eq. (1) by selecting appropriate parameters of the solutions. The comparison between our results and other's works are also given.     

A New Generalization of Extended Tanh-Function Method for Solving Nonlinear Evolution Equations

Making use of a new generalized ansatze and a proper transformation, we generalized the extended tanh-function method. Applying the generalized method with the aid of Maple, we consider some nonlinear evolution equations.As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extendedtanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain othernew and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profilesolitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.     

A General Mapping Approach and New Travelling Wave Solutions to (2+1)-Dimensional Boussinesq Equation

A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.     

Several Types of Similarity Solutions for the Hunter-Saxton Equation

We study separable and self-similar solutions to the HunterSaxton equation,a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal(among other applications).Essentially,we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the HunterSaxton equation.For each type of solution,we are able to obtain some simple exact solutions in closed-form,and more complicated solutions through an analytical approach.We find that there is a whole family of self-similar solutions,each of which depends on an arbitrary parameter.This parameter essentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data.The simpler solutions found constitute exact solutions to a nonlinear partial differential equation,and hence are also useful in a mathematical sense.Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions.Both types of solutions cast light on self-similar phenomenon arising in the HunterSaxton equation.     

A Series of Exact Solutions for a New (2+1)-Dimensional Calogero KdV Equation

An algebraic method is proposed to solve a new (2 1)-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.     

Solitary Wave and Doubly Periodic Wave Solutions to Three-Dimensional Nizhnik-Novikov-Veselov Equation

The generalized transformation method is utilized to solve three-dimensional Nizhnik-Novikov-Veselov equation and construct a series of new exact solutions including kink-shaped and bell-shaped soliton solutions, trigonometric function solutions, and Jacobi elliptic doubly periodic solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh methods and Jacobi function method, the method we used here gives more general exact solutions without much extra effort.     

(2+1)维色散长波方程的新的类孤子解

应用一种新的修改的代数方法去求解(2+1)维色散长波方程,获得方程的大量新的精确解.这些解包括类孤子解、类周期解、类有理解、类双曲函数解、类Jacobi椭圆函数解等等. 关键词： (2+1)维色散长波方程 类孤子解 类有理解 类双曲函数解 类Jacobi椭圆 函数解