一类分数阶非线性波方程的精确解及应用

Exact Solution for a Kind of Fractional Nonlinear Wave Equations and Its Application

基于分离变量的思想构造了分数阶非线性波方程含常系数的解的形式.在用待定系数法求解时,根据原方程确定假设解中的待定参数,得到具体解的表达式.利用该方法求解了3个非线性波方程,即分数阶CH(Camassa-Holm)方程、时间分数阶空间五阶Kdv-like方程、分数阶广义Ostrovsky方程.比较简便地得到了这些方程的精确解.文献中关于整数阶非线性波方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.对能够通过待定系数法求出精确解的分数阶微分方程所应满足的条件进行了阐述.

According to the characteristics of peaked soliton solution, the expression of exact solutions including undetermined coefficients is constructed for fractional nonlinear wave equations. The undetermined coefficient method is used. Determining the coefficients in the supposed solution according to the fractional nonlinear wave equations, the solution can be obtained. By means of the method several kinds of exact solutions are obtained for three nonlinear wave equations： the fractional Camassa-Holm, time fractional order and space fifth- order KdV-like, generalized fractional Ostrovsky. The corresponding three eact solutions are gained. The solutions given in literature about integer nonlinear wave equation become the special cases of the solutions in this paper. The graphs of some solutions are given through numerical simulation. The special conditions under which the fractional nonlinear wave equation will have exact solution are briefly described.