改进的截断展开法及其在变系数非线性方程中的应用

Improved Truncated Expansion Method and Its Applications to Variable Coefficent Nonlinearer Equations

本文对截断展开法进行了改进.首先,通过行波变换,将偏微分方程(PDE)转化为常微分方程(ODE).然后,在截断展开中,采用了非线性Riccati方程F′=p+qF+rF2将复杂的变系数非线性方程转变为一组超定代数方程组.再利用计算软件mathematic求解出代数方程组.从而得到变系数非线性演化方程的精确解.我们将这种方法应用于第一类变系数KdV方程和广义变系数KdV方程,得到了一系列精确解,其中包括一组Weierstrass椭圆函数解.这组解可以表示成Jacobi椭圆函数解,在模数m→1或m→0时这组解又可以分别退化为双曲函数解和三角函数解.

In this paper, the truncated expansion method is improved. First, with the aid of traveling wave transformation, partial differential equation（PDE） is reduced to an ordinary differential equation（ODE）. Then, in the truncated expansion method, using non-linear Riccati equation changes the problem solving an ordinary differential equation into another one solving the corresponding set of nonlinear algebraic equations with the help of computer algebraic system Mathematica. The method is applied to the first kind variable coefficient nonlinear equation and the general variable coefficient KdV equation and their kinds of exact solutions are obtained. Among these exact solutions, there is a kind of weierstrass elliptic function solution which expressed the corresponding Jacobi elliptic function solution. When modulus or, These solutions degenerate to the corresponding hyperbolic solutions and trigonometric function periodic solutions.