利用试探函数法求耦合KdV方程组的精确解

The Exact Solutions of the Coupled KdV Equations by Using Trial Function Method

通过引入一个变换和选择准确的试探函数,可以将非线性偏微分方程组化为一组易于求解的代数方程组,然后用待定系数法确定相应的系数,从而得到其精确解.将谢元喜(湖南理工学院学报:自然科学版,2011,24(4):12-15.)提出的试探函数进行改进,利用两种不同的试探函数,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组——耦合KdV方程组,从而得到了耦合KdV方程组的新显式精确解,其中包括一般形式的指数函数解、sech2型钟状正则孤波解和csch2型奇异行波解,此方法也可用于求其他非线性偏微分方程组的精确解.

Introducing a new transformation and selecting accurate trial function can convert nonlinear partial differential equations to a set of algebraic equations which can be easily solved,since its related coefficients are easily determined by the undetermined coefficients method.In this paper,the trial function in Y.X.Xie（J.Hunan Institute of Science and Technology： Natural Sci.,2011,24（4） ： 12-15.） was improved.As a result,some new explicit exact solutions of the well-known coupled KdV equations in nonlinear mathematical physics are obtained,which include the general form of exponential function solutions,sech2-type bell-profile regular solitary wave solution and csch2-type singular travelling wave solutions.This method can be applied to other nonlinear partial differential equations.