一类非线性偏微分方程Baecklund变换的分类

On Baecklund Transformations for a Class of Nonlinear Partial Differential Equation

讨论了形如u_t=u_(xxx)+F(u,u_x)的非线性偏微分方程由可积系统{v_x=P(v,u,u_t,u_x,u_(xx)),v_t=Q(v,u,u_t,u_x,u_(xx_)定义的Baecklund变换u→v分类问题,给出光滑函数F,P和Q的显式表达式,得到函数F只能是如下形式F(u,u_x)=cu_x^3+F_1(u)u_x,其中c是常数,并且分3种情形确定光滑函数F和相应的函数P和Q.

We classify nonlinear partial differential equations of the form ut = uxxx ＋ F （ u, ux ） which possess Baecklund transformations defined via associated integrable systems {vx=P（v,u,ut,ux,uxx）,vt=Q（v,u,ut,ux,uxx） We determine such smooth functions F, P and Q. Our results show that the function F can only be of the form F（u,vx）=cu^3x＋F1（u）ux where c is a constant and F1 is a smooth tunction.