一类高维非线性方程(组)自相似解的简易求法

A method for finding similarity solution of (2+1)-dimensional nonlinear partial differential equation(s)

对(2+1)维非线性偏微分方程进行相似变换后,根据相似变量不变性原理,提出了一个相似变量的复合变换,从而把(2+1)维偏微分方程最终化成常微分方程.将该方法用于KP方程、ZK方程、高维Burgers方程组,均得到了具有Palinlevé性质的常微分方程.通过进一步的分析求解得到KP方程和ZK方程的自相似渐进解,尤其是得到了高维耦合Burgers方程组的精确解.

Using a compounding transformation of similarity variables after similarity transformation, the Kadomfsov-Pefrishvili（KP） equation, the Zakharrov-Kuznetsov（ZK） equation and the higher-dimensional coupled Burgers equations are reduced to ordinary differential equations which are of Painlevé type. Furthermore, the approximate or asymptotical solutions of the KP equation and the ZK equation are obtained. Particully, the exact solution of the higher-dimensional coupled Burgers equation is obtained.