非线性系统的多线性分离变量法

Multi-linear variable separation approaches for nonlinear systems

线性物理中两大普遍适用的傅立叶变换法和分离变量法都不能直接应用到非线性物理,为此如何在非线性物理中建立相应的研究方法是众多物理学家和数学家们都非常关心的问题.本文介绍了一种适用于非线性系统的分离变量法—–多线性分离变量法,由其可以得到具有低维变量分离函数的多线性分离变量解.特别地,可积系统的多线性分离变量解通常包含有任意的低维变量分离函数.

It is known that the Fourier transform method and the variable separation approach,commonly applicable in linear physics,cannot be directly applied to nonlinear physics.Therefore,how to establish corresponding methods in nonlinear physics is a problem addressed by many physicists and mathematicians.In this paper,a variable separation method is presented,in which the multi-linear variable separation approach is applied to nonlinear systems to obtain multi-linear separation variable solutions with low-dimensional variable separation functions.The proposed approach is especially effective for the common case in which the multi-linear separation variable solution of an integrable system contains some arbitrary low-dimensional variable separation functions.