立方非线性Schrdinger方程的Weierstrass椭圆函数周期解

The new periodic wave solutions for cubic nonlinear Schrdinger equation

利用Weierstrass椭圆函数展开法对非线性光学、等离子体物理等许多系统中出现的立方非线性Schrdinger方程进行了研究.首先通过行波变换将方程化为一个常微分方程,再利用Weierstrass椭圆函数展开法思想将其化为一组超定代数方程组,通过解超定方程组,求得了含Weierstrass椭圆函数的周期解,以及对应的Jacobi椭圆函数解和极限情况下退化的孤波解.该方法有以下两个特点:一是可以借助数学软件Mathematica自动地完成;二是可以用于求解其它的非线性演化方程(方程组).

The cubic nonlinear Schroedinger（CNLS）equation, which＇exist widely in some systems such as plasma physics, nonlinear optics etc, has been studied by using the Weierstrass elliptic function expansion method. With the aid of travelling wave transformation, the CNLS equation can be reduced to an ordinary differential equation. Then, the main step of this algorithm changes the problem solving an ordinary differential equation into another one solving the corresponding set of nonlinear algebraic equations. As a conclusion, some new doubly periodic wave solutions are obtained in terms of the Weierstrass elliptic function. Meariwhile, the corresponding Jacobi elliptic function solutions and the solitary wave solutions are derived in the limit case. The method has two virtues： one is the process can be performed in computerized symbolic computation system such as Mathematica. Another is the method can be also applied to many nonlinear differential equation（equations） in mathematical physics.