Study on Exact Analytical Solutions for Two Systems of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｒｉｎｇｔｈｅｃｏｕｒｓｅｏｆｓｔｕｄｙｉｎｇｔｈｅｗａｔｅｒｗａｖｅ,ｍａｎｙｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｍｏｄｅｌｓｗｅｒｅｏｂｔａｉｎｅｄ ,ｓｕｃｈａｓＫｄＶｅｑｕａｔｉｏｎ ,ｍＫｄＶｅｑｕａｔｉｏｎ ,(2 1 )＿ｄｉｍｅｎｓｉｏｎａｌＫＰｅｑｕａｔｉｏｎ ,ｃｏｕｐｌｅｄＫｄＶｅｑｕａｔｉｏｎｓ,ｖａｒｉａｎｔＢｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎｓ ,ＷＫＢｅｑｕａｔｉｏｎｓｅｔｃ .[1- 13 ].Ｉｎｏｒｄｅｒｔｏｆｉｎｄｅｘｐｌｉｔｉｃｅｘａｃｔｓｏｌｕｔｉｏ…     

Study of Explicit Analytic Solutions for the Nonlinear Coupled Scalar Field Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅ,Ｍａｎｙｐｈｅｎｏｍｅｎａｉｎｐｈｙｓｉｃｓ,ｍｅｃｈａｎｉｃｓ,ｃｈｅｍｉｓｔｒｙａｎｄｂｉｏｌｏｇｙｅｔｃ.ｃａｎｂｅｄｅｓｃｒｉｂｅｄｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｙｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌ＿ｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎｓ[1- 7].Ｏｎｔｈｅｃｏｎｔｒａｒｙ ,ｉｎｏｒｄｅｒｔｏｓｔｕｄｙｔｈｅｓｅｐｈｅｎｏｍｅｎａｑｕａｎｔｉｔａｔｉｖｅｌｙ .Ｉｔｉｓｖｅｒｙｉｍ…     

Auto-Darboux Transformation and Exact Solutions of the Brusselator Reaction Diffusion Model

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＢｒｕｓｓｅｌａｔｏｒｒｅａｃｔｉｏｎｍｏｄｅｌｐｌａｙｓａｎｉｍｐｏｒｔａｎｔｒｏｌｅｂｏｔｈｉｎｂｉｏｌｏｇｙａｎｄｉｎｃｈｅｍｉｓｔｒｙ .ＳｉｎｃｅｔｈｅｍｏｄｅｌｗａｓｐｕｔｆｏｒｗａｒｄｂｙＰｒｉｇｏｇｉｎｅａｎｄＬｅｆｅｖｅｒｉｎ 1 968,ｍｕｃｈａｔｔｅｎｔｉｏｎｈａｄｂｅｅｎｐａｉｄｔｏｔｈｅｍｏｄｅｌａｎｄｍａｎｙｐｒｏｐｅｒｔｉｅｓｏｆｉｔｈａｄｂｅｅｎｒｅｓｅａｒｃｈｅｄｂｙｍａｎｙｐｅｏｐｌｅｖｉａｕｓｉｎｇｄｉｆｆｅｒｅｎｔｍｅｔｈｏｄｓ[1- 5…     

（２＋１）维非线性色散长波方程的相似约化和解析解

首先借助于Ｍａｔｈｅｍａｔｉｃａ软件,将Ｃｌａｒｋｓｏｎ和Ｋｒｕｓｋａｌ引入的直接约化法推广并应用于（２＋１） 维偏微分方程组情形 （２＋１） 维非线性色散长波方程,获得了该方程的六种类型的相似约化和若干解析解,其中包括ＰａｉｎｌｅｖｅⅡ型方程和孤子解．然后基于文［５］的结论,通过引入新的级数变换,获得了该方程的有理分式解析解．这种方法也适合于其它的微分方程．     

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.     

分数外微分在三维空间中的应用

首先介绍了分数微积分和分数微分形式。讨论了在原点处对曲线坐标的分数外微分变换,并且获得了从三维卡氏坐标到球面坐标和柱面坐标的两个分数微分变换。特别地,当v=m=1时,这两个分数微分变换约化的结果与通过外微积分获得的结果是一致的。     

获得非线性微分方程显式解析解的两种新算法

基于AC=BD的思想来求解非线性微分方程(组)。设Au=0为给定的待求解的方程,Dv=0是容易求解的方程。如果可以获得变换u=Cv使得v满足Dv=0,则能够得到Au=0的解。为了说明该种途径,本文举例给出几种变换C的表达式。     

NEW B$\ddot{A}$CKLUND TRANSFORMATION AND EXACT SOLUTIONS FOR VARIABLE COEFFICIENT KdV EQUATION

In this paper, with the aid of Lax pairs, a new B?cklund transformation for the variable coefficient KdV equation is found, Based on the B?cklund transformation, only if integration is needed, a series of exact solutions can be obtained. This method is important for finding more new and physical-signficant solutions.