四阶变系数非线性偏微分方程的Bcklund变换

Bcklund transformations for the 4th-order nonlinear partial differential equations with variable coefficients

利用Bcklund变换u→v及其所满足的可积系统vx=P(x,t,v,u,ux,uxx,uxxx),vt=Q(x,t,v,u,ux,uxx,uxxx),将四阶变系数非线性偏微分方程ut=F(x,t,u,ux,uxx,uxxx,uxxxx)进行分类.同时,利用分类结果给出一些方程的有理函数解.该方法也适用于高阶非线性偏微分方程.

This paper classifies the fourth-order nonlinear partial differential equations with variable coefficient of the form ut=F（x,t,u,ux,uxx,uxxx,uxxxx） which possess Bcklund transformations u→v defined via associated integrable systems of the form vx=P（x,t,v,u,ux,uxx,uxxx）,vt=Q（x,t,v,u,ux,uxx,uxxx）.Some rational function solutions with the classification are obtained.The method also can be applied to the higher order equations.