一类二阶非线性偏微分方程B(a)ecklund变换的分类

摘要

讨论形如ut=F(u,ux,uxx)的非线性偏微分方程由可积系统vx=P(v,u,ux),vt=Q(v,u,ux)定义的B(a)cklund变换u→v分类问题,证明了这样的非线性偏微分方程只能是Burgers方程ut=uxx+2uux,而相应的可积系统是vx=(λ+v)(u-v),vt=(λ+v)(u2+ux-uv)-λ(λ+v)(u-v),其中λ是任意常数.%Classify B(a)cklund transformations u ι→v for partial differential equations of the form ut= F(u, ux, uxx )which are defined via associated integrable systems of the form vx = P ( v, u, ux ), vt = Q( v, u, ux ) is classified. It is showed that the only such nonlinear partial differential equation is the Burgers' equation ut = uxx + 2uux, and the associated integra ble system is vx = (λ,+ v) (u - v), vt = (λ, + v) (u2 + ux - uv) - λ (λ + v) ( u - v), where λ is an arbitrary constant.