低维非线性系统的一般多线性变量分离方法和局域激发模式

基于Bcklund变换的多线性变量分离方法(BT-MLVSA)是求解非线性系统的一种非常有效的方法. 一般多线性变量分离方法(GMLVSA)是该方法的推广. 实现GMLVSA主要有四种途径，一是先把场量按照多个任意函数(通常考虑两个函数的情形)展开得到关于多个函数的多线性方程，另一种途径是推广变量分离的假设，第三类是基于Darboux变换的多线性变量分离方法(DT-MLVSA)，第四类是导数相关泛函变量分离法. 利用第一类GMLVSA，可以得到(2+1)维mNNV系统和sine-Gordon系统的一般多线性变量分离解. 把第一类GMLVSA推广到二维非线性系统，这些系统是通过对称约化(2+1)维sine-Gordon.系统得到的. 也就是说，一般多线性变量分离可解性在对称约化下从高维系统到低维系统得到了保持. 这也提供了一条从高维非线性系统导出可GMLVSA求解的低维非线性系统的有效途径. 关键词：Bcklund变换多线性变量分离sine-Gordon系统对称约化

General multi-linear variable separation approach to solving low dimensional nonlinear systems and localized exitations

Multi-linear variable separation approach based on the corresponding Bcklund transformation (BT-MLVSA) is a useful method to solve nonlinear systems. General multi-linear variable separation approach (GMLVSA) is its extension and there are four ways to realize it. The first one is to expand the nonlinear systems according to multi-arbitrary functions, the second one is to expand the variable separation ansatz. The third one is the MLVSA based on the Darboux transformation (DT-MLVSA) and the last one is the derivative-dependent functional variable separation method. By using the first kind of GMLVSA, the solutions can be obtained for the (2+1)-dimensional mNNV system and sine-Gordon system. In this paper, the first kind of GMLVSA is extended to solve some two-dimensional nonlinear systems which are derived from the (2+1)-dimensional sine-Gordon system by using symmetry reduction method. Namely, the applicability of the method is retained from high dimensional systems to low dimensional systems in the symmery reduction sense. This also provide a way of deducing low dimensional systems which can be solved by GMLVSA from high dimensional systems.