扰动变系数组合KdV方程的同伦映射解

Homotopic mapping solutions for variable coefficient combined perturbed KdV equation

利用同伦映射法求解了扰动变系数组合KdV方程双周期形式的近似解.首先通过一个函数变换将所要研究的扰动变系数组合KdV方程简化为扰动常系数组合KdV方程,然后引入一个同伦映射,通过傅里叶分析等手段求出原方程在给定初始条件下的近似解析解,主要是Jacobi椭圆函数形式的近似解.这些解在极限情形下有的可退化为双曲函数形式的近似解,有的可退化为三角函数形式的近似解,有的存在2种形式的近似解.最后给出了在微扰情形下变系数组合KdV方程的一次近似解和二次近似解.

The homotopic mapping method was used to obtain the approximate solutions with double periodic form of coefficienl combined perturbed KdV equations. By functional transformation, the variable coefficient combined perturbed KdV equations were simplified to ordinary line array and combined perturbed KdV equations. Based on Fourier analysis method, the homotopy mapping was introduced to get approximate solutions of Jacobi elliptic function form for original equations under initial conditions. Some solutions could be degenerated to approximate solutions of hyperbolic function form or trigonometric function form in the limit cases. The first approximate solutions and the second approximate solutions of varia- ble coefficient combined KdV equations were obtained under perturbation condition.