非线性 Schrödinger 方程的保辛数值求解

摘要

首先将非线性 Schrödinger 方程化为 Hamilton 正则方程形式，而后建立 Hamilton 体系下的变分原理。再用有限元法离散空间坐标，同时对时间坐标进行精细积分，最后运用混合能变分原理，提出非线性 Schrödinger 方程保辛数值解法。这种解法在保辛的同时，可以让能量和质量在积分格点上亦全部达到守恒。数值算例验证了该方法的有效性。%This paper proposes a new numerical method with symplectic preserving to nonlinear Schrödinger equation,and the validity of this method is proved by numerical examples.We firstly trans-form nonlinear Schrödinger equation to Hamilton equations and therefore found Hamilton variational principle,followed with the discrete space coordinate through finite element method,precise integration algorithm used on time coordinate,and then with the mixed-energy variational principle,a numerical symplectic-preserving solution of nonlinear Schrödinger equation in the paper is well presented,while energy and mass preserving is realized simultaneously on the integration grids.Numerical examples later on demonstrate the effectiveness of this method.