用改进的试探函数法求解广义KdV方程

试探函数法求解非线性数学物理中一个非常著名的非线性偏微分方程—广义KdV方程,求得其一般形式的指数函数解,据此不但求得了广义KdV方程的sech2型钟状正则孤波解,而且求得了其csch2型奇异行波解,最后,利用一些熟知的数学关系式,又求得其若干其它显式精确解,包括三角函数型周期波解等。

Solving the Generalized KdV Equation by Applying the Improved Trial Function Method

We make two distinct and different modifications of the trial function method put forward in Ref.As a consequence,by taking full advantage of this improved trial function method,we may find the more general solution of the exponential function form for the well-known generalized KdV equation in nonlinear mathematical physics in a concise and direct manner.On the basis of it,not only the sech2-type bell-profile regular solitary wave solution but also the csch2-type singular travelling wave solution with importantly physical significance of the generalized KdV equation is able to be readily found.At last,with the bell-profile some well-known mathematical formulae,we may still find a lot of other explicit and exact solutions to the generalized KdV equation,which include trigonometric function periodic wave solutions and the like.