用试探函数法求KdV方程的孤子解

Search for the Soliton Solution to the KdV Equationby Means of Trial Function Method

通过引入一个新的变换,利用试探函数法,并选取准确的试探函数形式,将一个难于求解的非线性偏微分方程化成了一组易于求解的非线性代数方程,从而简洁地求得了KdV方程的孤子解,所得结果与已有结果完全吻合。这种方法可望进一步推广用于求解其它非线性偏微分方程。

By using trial function method and introducing a new transformation, the nonlinear partial differential equation that is hard to be solved by making use of the regular technique can be reduced to a set of nonlinear algebraic equations, which can be easily solved, and their related coefficients can be easily determined by virtue of taking advantage of the approach of undetermined coefficients. Finally, the explicit exact soliton solution to the KdV equation is successfully derived. The results are in good agreement with those obtained in some existing references. It is not difficult to see that this method used herein is particularly simple and concise. We firmly believe that this approach used herein may be generalized to construct the solutions to other nonlinear partial differential equations.