用双曲函数法求KdV-mKdV方程的钟状孤波解和激波状孤波解

提出一种统一的求解非线性演化方程孤波解的双曲函数法，并利用这种方法求出了组合KdV-mKdV方程的钟状孤波解和激波状孤波解．作为特例，可以给出mKdV方程的两类孤波解，而且还给出了KdV方程的钟状孤波解．双曲函数法是利用非线性波动方程孤波解的局部性特点，将方程的孤波解表示为双曲函数的多项式，从而将非线性波动方程的求解问题转化为非线性代数方程组的求解问题．因此双曲函数法是一种简单而实用的方法．

Two Kinds of Solitary Wave Solutions to the KdV-mKdV Equation Obtained by Hyperbolic Function Method

A united hyperbolic function method to find the solita ry wave solutions to nonlinear evolution equations was proposed,and two kinds of solitary wave solutions to the combined KdV-mKdV equation were obtained by this method.As a special example,two kinds of solitary wave solutions to the mKdV equation can be obtained,and the bell-shaped solution to the KdV equation was also given.The proposed method is based on the fact that the solitary wave solutions are essentially of a localized nature.In this method,the solitary wave solutions to a nonlinear wave equation are denoted as the polynomials of hyperbolic functions,and the nonlinear wave equation is changed into nonlinear algebraic equations.So the hyperbolic function method is simple and effective when used to study the solitary wave solutions of the nonlinear evolution equation.