Bcklund变换下一类Burgers方程精确解分析

Exact Solution of Burgers’Equation for Bcklund Transformations

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cklund变换u→v分类,其最简Burgers方程为ut=uxx+2uux,相应的可积系统是vx=(λ+v)(u-v),vt=(λ+v)(u2-ux-uv)-λ(λ+v)(v-v),其中,λ是任意常数。将Bcklund变换连续n次作用于Burgers方程的零解u0(x,t)≡0,并且每次取不同的参数λk(1≤k≤n),得到了Burgers方程的精确解un(x,t),并揭示了Burgers方程光滑和(或)奇异扭结解相互作用的过程。

Bcklund transformations u→v for partial differential equations of the form ut=F(u,ux,uxx) are defined via associated integrable systems of the form vx=P(v,u,ux),vt=Q(v,u,ux), only such nonlinear partial differential equation is the Burgers’ equation ut=uxx+2uux, and the associated integrable system.is vx=(λ+v)(u-v),vt=(λ+v)(u2+ux-uv)-λ(λ+v)(u-v), where λ is an arbitrary constant.It repeats the above Bcklund transformation to the zero solution u0(x,t)=0 of the Burgers’ equation, and taking differential λk(1≤k≤n) ie used to get a lot of new exact solutions of the Burgers’ equation.All these solutions reveal the interaction process of smooth and/or singular kink solutions of the Burgers’ equation.