一类非线性演化方程新的行波解

利用双函数法和吴消元法,得到了一类非线性演化方程在不同情况下的一系列显示精确解.Sinh-Gordon方程及Klein-Gordon方程作为该方程的特例也得到了相应的行波解.     

应用改进的简单方程法求非线性数学物理方程的精确解

应用改进的简单方程法求得Cahn-Allen方程和Jimbo-Miwa方程的精确解,这些解包括双曲函数解、三角函数解.当对双曲函数解中的参数取特殊值时,可以得到了孤立波解.当对三角函数解中的参数取特殊值时,可以得到对应的周期波函数解.实践证明,简单方程法对于研究非线性数学物理方程具有非常广泛的应用意义.     

广义变形的DP-CH方程的显式精确解

通过构造一个辅助方程,利用动力系统方法,Jacobi椭圆函数,来讨论广义变形的DP-CH方程,得到了广义变形的DP-CH方程更丰富的行波解的显示精确表示.     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

浅水波方程和Klein-Gordon方程新的精确行波解

采用了一种新的方法来求解浅水波方程和Klein-Gordon的行波解.在该方法下,Klein-Gordon方程和浅水波方程都得到了其精确的周期孤立波解,从而该方法的有效性得到了验证.     

RLW-Burgers方程的一类解析解

本文给出了 RLW-Burgers方程及 Kd V-Burgers方程的一类解析解 ,且可得到 RLW-Burgers方程的振荡激波解 .这些解可以表示为 Burgers方程和 Kd V方程解的线性组合 ,文末还对文 [8]作了讨论 .     

非齐次Burgers方程解的渐近性行为

构造了非齐次Burgers方程的解,方程服从有界和紧致的初始曲线[Kloosterziel RC.J Engrg Math,1990,24(3):213-236],作了一个有趣的探索.将热方程初值问题(L2(R,ex2/2)中有初值)的解,表示为该热方程自相似解的一个级数,Kloosterziel方法立即显示出该初值问题解的渐近性行为.受Kloosterziel方法的启发,根据热方程的自相似解,来表示非齐次Burgers方程的解.最后得到该非齐次Burgers方程解的渐近性特征.     

广义Pell方程及其应用

本文给出了广义Ｐｅｌｌ方程的一切有理数解公式，应用它得到了一类Ｌｅｇｅｎｄｒｅ方程的一切整数解公式．     

ODE方程精确解及其应用

本文得到了一类ＯＤＥ方程精确解,并给出了它在Ｃｈａｆｆａ－Ｉｎｆａｎｔｅ方程和波方程上的应用。     

CHANDRASEKHAR H—方程的投影解法

本文讨论一类非线性积分方程——Chandrasekhar H-方程的投影解法,其方法是将方程化为压缩型算子方程的投影解法。在适当条件下,得到投影方程的解的存在唯一性并且投影解收敛于原方程的解。     

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.     

On Nonlinear Hyperbolic Functional Differential Equations

In this paper, existence of weak solutions of second order evolution equations is proved and some properties of the solutions are shown. The results are applied to higher order nonlinear hyperbolic functional differential equations.     

Quasi-rational function solutions of an elliptic equation and its application to solve some nonlinear evolution equations

Using the differential transformation method and the homogeneous balance method, some new solutions of an auxiliary elliptic equation are obtained. These solutions possess the forms of rational functions in terms of trigonometric functions, hyperbolic functions, exponential functions, power functions, elliptic functions and their operation and composite functions and so on, which are so-called quasi-rational function solutions. Based on these new quasi-rational functions solutions, a direct method is proposed to construct the exact solutions of some nonlinear evolution equations with the aid of symbolic computation. The coupled KdV-mKdV equation and Broer-Kaup equations are chosen to illustrate the effectiveness and convenience of the suggested method for obtaining quasi-rational function solutions of nonlinear evolution equations.     

On reduction of some classes of partial differential equations to equations with fewer variables and exact solutions

We establish a connection between the fundamental solutions to some classes of linear nonstationary partial differential equations and the fundamental solutions to other nonstationary equations with fewer variables. In particular, reduction enables us to obtain exact formulas for the fundamental solutions of some spatial nonstationary equations of mathematical physics (for example, the Kadomtsev-Petviashvili equation, the Kelvin-Voigt equation, etc.) from the available fundamental solutions to one-dimensional stationary equations.     

EXACT SOLUTIONS OF THE VARIABLE COEFFICIENT KdV AND SG TYPE EQUATIONS

In this paper,the variable cofficient KdV equation with dissipative loss and nonuniformity terms and the variable coefficient SG equation with nonuniformity term are studied. The exact solutions of the KdV and SG equations are obtained. In particular,the soliton solutions oftwo equations are found.     

Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order

In this paper, we present several methods of judging shape of the solitary wave and solution formulae for some nonlinear evolution equations by means of Lienard equations. Then, using the judgement methods and solution formulae, we obtain solutions of the solitary wave for some of important nonlinear evolution equations, which include generalized modified Boussinesq, generalized nonlinear wave, generalized Fisher, generalized Klein-Gordon and generalized Zakharov equations. Some new solitary-wave solutions are found for the equations.     

Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay

In this paper, we generalize some integral inequalities to more general situations, and the inequalities of Pachpatte type are corollaries of our's. We establish bounds on the solutions, and we show the usefulness of our results in investigating the asymptotic behavior and the stability on the solutions of integral equations, differential equations and integro-differential equations with time delay.     

Nonautonomous first integrals and general solutions of the KdV‐Burgers and mKdV‐Burgers equations with the source

The method for constructing first integrals and general solutions of nonlinear ordinary differential equations is presented. The method is based on index accounting of the Fuchs indices, which appeared during the Painlevé test of a nonlinear differential equation. The Fuchs indices indicate us the leading members of the first integrals for the origin differential equation. Taking into account the values of the Fuchs indices, we can construct the auxiliary equation, which allows to look for the first integrals of nonlinear differential equations. The method is used to obtain the first integrals and general solutions of the KdV‐Burgers and the mKdV‐Burgers equations with a source. The nonautonomous first integrals in the polynomials form are found. The general solutions of these nonlinear differential equations under at some additional conditions on the parameters of differential equations are also obtained. Illustrations of some solutions of the KdV‐Burgers and the mKdV‐Burgers are given.     

Oscillatory properties for semilinear degenerate hyperbolic equations of second order

We consider three kind of oscillatory properties of the solutions to semilinear degenerate hyperbolic equations. Several sufficient conditions for the oscillation or non-oscillation are presented. In particular, they give us the positivity of the solutions for semilinear hyperbolic equations degenerating at initial point in one space dimension. Moreover we establish a few oscillatory conditions for the solutions of the mixed problem reduced to in one space dimension.