用改进的试探函数法求解Boussinesq方程

Solving the Boussinesq Equation by Means of the Improved Trial Function Method

对文献《用试探函数法求KdV方程的孤子解》中所提出的试探函数法进行了两点明显的改进,并把它用于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-Boussinesq方程,从而简洁地求得了其一般形式的指数函数解,据此不但求得了Boussinesq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,又求得了其若干其它显式精确解,包括三角函数型周期波解等,所得结果包含了已有的结果和一些新的或更一般的结果。本方法可望进一步推广用于求解非线性数学物理中的其它非线性偏微分方程。

In the present paper, we make two distinct modifications of the trial function method put forward in Ref. [ 23 ]. As a consequence, by making use of this improved trial function method, we may find the more general solution of the exponential function form to the well - known Boussinesq equation in nonlinear mathematical physics in a concise and direct manner. On the basis of it, not only the seeh2 - type bell - profile regular solitary wave solution but also the csch2 - type singular travelling wave solution with importantly physical significance for the Boussinesq equation is able to be readily found. At last, with the aid of some well - known mathematical formulae, we may still find a number of other explicit and exact solutions of the Boussinesq equation, which comprise trigonometric function periodic wave solutions and so on. The results obtained herein contain the existing ones given in other lots of previous literatures and some new or more general ones which can not be seen in the existing literatures to the best of our knowledge. We firmly believe that this approach used herein may be generalized to look for the explicit and exact solutions of other nonlinear partial differential equations in nonlinear mathematical physics.