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

Abstract

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.