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.

Generalized conditional symmetries and derivative-dependent functional variable separation approaches for nonlinear systems

It is well known that both the Fourier analysis and the variable separation approach are effective means for solving problems in linear systems. However, it is quite difficult to extend the variable separation approach into nonlinear systems. It is worth the effort to develop new approaches of variable separation to deal with some nonlinear systems. In this paper, we bring forward a novel concept of the derivative-dependent functional separable solutions (DDFSSs). Under the given definition, its corresponding generalized conditional symmetry (GCS) is sought, and elementary theories involving the DDFSS approach is established. Next, the DDFSS approach is applied to the generalized nonlinear diffusion equations, the KdV type equations and the extended nonlinear wave equations, respectively. The following objectives have been achieved in this study: (1) a complete list of canonical forms is acquired which expresses the classification results of the considered equations; (2) the exact solutions are obtained to the resulting equations via the GCS method; (3) the localized excitations of certain solutions are characterized and the symmetry group interpretation is given to some solutions. Finally, some points can be made for the fact that the DDFSS approach can be developed to solve some high dimensional equations, systems of equations, and can also be extended even further to deal with some perturbed nonlinear systems.