二维非线性Schrdinger方程的辛与多辛格式

SYMPLECTIC AND MULTI-SYMPLECTIC SCHEMES FOR THE TWO-DIMENSIONAL NONLINEAR SCHRDINGER EQUATION

对满足周期边界条件的二维非线性Schroedinger方程，运用中心差分对该方程进行空间离散，得到一个有限维Hamilton系统，然后用隐式Euler中点格式进行时间离散得到其辛格式，针对该方程的多辛形式，运用有限体积法离散， 得到一种直平行六面休上的中点型多辛格式．用所构造的辛与多辛格式对二维非线性Schroedinger方程的平面波解和奇异解进行数值模拟，结果验证了所构造格式的有效性．最后，根据计算结果，对两种格式进行，分析和比较．

The two-dimensional nonlinear Schroedinger equation （2D NLSE） with periodic boundary condition is considered in this paper. An implicit symplectic scheme is constructed by using central difference scheme in space and implicit Euler-centered scheme in time. In addition, a midpoint rule multi-symplectic method is obtained by applying a cell vertex finite volume discretization to its multi-symplectic form. Numerical simulations are presented for plane wave solution and singular solution of the 2D NLSE. The results demonstrate the effectiveness of the proposed methods. Furthermore, the two methods are analyzed and compared with each other.