某些偏微分方程的定向振荡解

Oriented Oscillatory Solutions to Some Partial Differential Equations

本文研究了在应用中颇为重要的几类非线性偏微分方程的振荡解。首先,我们讨论了修正KdV方程、二维KdV方程和Boussinesq方程,利用Jacobl椭圆函数作出了这些方程转化后的常微分方程的解,从而证明了原方程行波振荡解的存在性。其次,我们研究了高维约比波动方程。对所归结的微分方程构造了它的一个幂级数解,导出了此解与Bessel函数的关系,然后由Bessel函数的实零点的分布结果证明了高维约化波动方程的柱面振荡解的存在性。

This paper discusses the oscillatory solutions to some partial differential equations.First,for the modified KdV equation,the two- -dimensional KdV equation and Boussinesq equations,the existence of travelling wave oscillatory solutions are proved by using the Jacobi elliptic functions.Then the higher dimensional reduced wave equation is investigated, and an explicit solution is constructed in terms of the Bessel functions. Finally,it is shown that,by using some results of the zeros of the Bessel functions,the solution is a cylindrical oscillatory one.