2+1 维变系数广义KP方程的椭圆周期解

运用Jacobi椭圆函数展开法求得了2 1维变系数广义KadoratsevPetviashvili方程的椭圆周期解及孤立波解．     

含变系数或强迫项的KdV方程的新解

Jacobi椭圆函数展开法被推广并用于求解另一种形式的KdV方程的新的精确解,所求解的这类KdV方程包括一种典型的变系数的KdV方程和具有强迫项（随机项）的KdV方程．用这种方法得到的新的类周期解在极限条件下可以退化为类孤立波解或类冲击波解．     

扩展浅水波方程的精确相互作用解

将相容的双曲正切函数展开法(CTE方法)和截断Painlevé分析法应用于扩展浅水波方程,并通过这两个方法求解相容性方程的若干精确相互作用解,包括如孤子与周期波相互作用解、变振幅周期波与椭圆周期波相互作用解.     

应用F展开法求KdV方程的周期波解

提出了求非线性数学物理演化方程周期波解的F展开法,该方法可看作最近提出的扩展的Jacobi椭圆函数展开方法的浓缩.直接利用F展开法而不计算Jacobi椭圆函数,我们可同时得到著名的KdV方程的多个用Jacobi椭圆函数表示的周期波解.当模数m→1 时,可得到双曲函数解(包括孤立波解).     

广义随机KP方程的椭圆周期解

利用Hermite变换和Jacobi椭圆函数展开法研究(2+1)-维广义随机Kadomtsev-Petviashvili方程,并给出了它的随机椭圆周期解及随机孤立波解.     

耦合Schrdinger-Boussinesq方程组的显式精确解

本文针对耦合Schrodinger-Boussinesq方程组,借助于F-展开法得到了用不同Jacobi椭圆函数表示的一系列周期波解．在极限情况下,还求出了对应的孤立波解．     

耦合Schr(o)dinger-Boussinesq方程组的显式精确解

本文针对耦合Schrodinger-Boussinesq方程组，借助于F-展开法得到了用不同Jacobi椭圆函数表示的一系列周期波解．在极限情况下，还求出了对应的孤立波解．     

Klein-Gordon-Zakharov方程组的周期波解

通过使用符号计算系统Mathematica,并借助于推广的F-展开法,我们得到了Klein- Gordon-Zakharov方程组的用不同Jacobi椭圆函数表示的一系列周期波解．在极限情况下,还求出了对应的孤立波解．     

Davey StewartsonⅠ的周期波解

利用新近提出的F展开法，导出了Davey StewartsonⅠ方程的由Jacobi椭圆函数表示的周期波解；并且在极限的情况下，得到了Davey StewartsonⅠ方程的孤波解以及其他形式解.     

寻找具有三个任意函数的变系数KdV-MKdV方程的类孤波解的新方法

给出了求具有三个任意函数的变系数非线性演化方程的类孤波解的截断展开方法.这种方法的关键是首先把形式解设为几个待定函数的截断展开形式,从而可将变系数非线性演化方程转化为一组待定函数的代数方程,然后进一步给出容易积分的待定函数的常微分方程组,从而构造出相应的类孤波解.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

KK方程和改进的Boussinesq方程的新精确解

应用改进的Fan's代数方法,得到了KK方程和改进的Boussinesq方程的一系列新精确解,包括孤立波解、类孤立波解、纽结波解、奇异纽结波解和三角函数周期解.     

EXACT TRAVELING WAVE SOLUTIONS OF MODIFIED ZAKHAROV EQUATIONS FOR PLASMAS WITH A QUANTUM CORRECTION＊

In this article,the authors study the exact traveling wave solutions of modified Zakharov equations for plasmas with a quantum correction by hyperbolic tangent function expansion method,hyperbolic seca...     

Discrete exact solutions of modified Volterra and Volterra lattice equations via the new discrete sine-Gordon expansion algorithm

Recently, we have presented a sine-Gordon expansion method to construct new exact solutions of a wide of continuous nonlinear evolution equations. In this paper we further develop the method to be the discrete sine-Gordon expansion method in nonlinear differential-difference equations, in particular, discrete soliton equations. We choose the modified Volterra lattice and Volterra lattice equation to illustrate the new method such as many types of new exact solutions are obtained. Moreover some figures display the profiles of the obtained solutions. Our method can be also applied to other discrete soliton equations.     

The exponential function rational expansion method and exact solutions to nonlinear lattice equations system

We propose an exponential function rational expansion method for solving exact traveling wave solutions to nonlinear differential-difference equations system. By this method, we obtain some exact traveling wave solutions to the relativistic Toda lattice equations system and discuss the significance of these solutions. Finally, we give an open problem.     

New Explicit and Exact Solutions for the Klein-Gordon-Zakharov Equations

In this paper, based on the generalized Jacobi elliptic function expansion method, we obtain abundant new explicit and exact solutions of the Klein-Gordon- Zakharov equations, which degenerate to solitary wave solutions and triangle function solutions in the limit cases, showing that this new method is more powerful to seek exact solutions of nonlinear partial differential equations in mathematical physics.     

Group-invariant solutions of the Fokker-Planck equation

Group-invariance under infinitesimal transformations is used to generate a wide class of solutions of some Fokker-Planck equations. The partial differential equation in two variables is reduced to an ordinary differential equation; reduction of the latter to standard forms is noted in most cases. Some of the known existing solutions are obtained as particular cases. Only self-similar types of solutions are discussed. The appearance of a free parameter that can be treated as an eigenvalue (or transform variable) offers flexibility in constructing new solutions. Some solutions of this parabolic equation have wave-like features. The general results can also be used to solve some types of moving-boundary problems.     

A series of the solutions for the Heisenberg ferromagnetic spin chain equation

The Heisenberg ferromagnetic spin chain equation is investigated. By applying the improved F‐expansion method (Exp‐function method) and the Jacobi elliptic method, respectively, a series of exact solutions is constructed. The parametric conditions of the existence for the solutions are presented. These solutions comprise periodic wave solutions, doubly periodic wave solutions, and dark and bright soliton solutions, which are expressed in several different function forms, namely, Jacobi elliptic function, trigonometric function, hyperbolic function, and exponential function. The results illustrate that the Exp‐function method is a powerful symbolic algorithm to look for new solutions for the nonlinear evolution systems.     

新的双曲函数有理展开法和符号计算构造差分mkdv的孤波解

本文通过利用一个广泛的题设提出一种推广的双曲函数展开法, 并利用此方法求解了 离散的 mKdV 方程, 获得了丰富的显式精确解. 此方法可以用于求其他非线性系统的精确解.     

(2+1)维Burgers方程新的复合解

In this paper, a new generalized compound Riccati equations rational expansion method （GCRERE） is proposed. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method is not only recover some known solutions, but also find some new and general complexiton solutions. Being concise and straightforward, it is applied to the （2＋1）-dimensional Burgers equation. As a result, eight families of new exact analytical solutions for this equation are found. The method can also be applied to other nonlinear partial differential equations.