Variable Separation and Derivative-Dependent Functional Separable Solutions to Generalized Nonlinear Wave Equations

Using the generalized conditional symmetry approach, we obtain a number of new generalized (1+1)-dimensional nonlinear wave equations that admit derivative-dependent functional separable solutions.

Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source

By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.

Approximate Derivative-Dependent Functional Variable Separation for the Generalized Diffusion Equations with Perturbation

As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.

Variable Separation for(1+1)-Dimensional Nonlinear Evolution Equations with Mixed Partial Derivatives

We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.

非线性系统的一般条件对称和导数相关泛函分离变量法

众所周知,Fourier分析和分离变量法是研究线性系统的有效手段,但分离变量法难以推广到非线性系统.发展处理非线性系统的新的分离变量法迫在眉睫.首先提出导数相关泛函分离变量解(DDFSS)的新概念,给出定义,据此寻求相应的一般条件对称(GCS),创建理论体系(简称DDFSS方法).尔后,分别应用DDFSS方法于一般非线性扩散方程、KdV类方程、一般非线性波动方程:(1)对所考察方程做了DDFSS可解的完全归类;(2)利用DDFSS方法建立了所得分类方程的DDFSS精确解;(3)描述了一些解的局域激发等性质,给出了有关结果的对称群解释.最后指出,DDFSS方法可发展应用于求解某些高维方程和方程组问题,更能扩展用于处理带扰动项的非线性系统.