MULTI-SYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHR ¨ ODINGER EQUATION

Abstract

Bridges and Reich suggested the idea of multi-symplectic spectral discretization on Fourier space (4). Based on their theory, we investigate the multi-symplectic Fourier pseudospectral discretization of the nonlinear Schr¨ odinger equation (NLS) on real space. We show that the multi-symplectic semi-discretization of the nonlinear Schr¨ odinger equation with periodic boundary conditions has N (the number of the nodes) semi-discrete multi- symplectic conservation laws. The symplectic discretization in time of the semi-discretization leads to N full- discrete multi-symplectic conservation laws. We also prove a result relating to the spectral differentiation matrix. Numerical experiments are included to demonstrate the remarkable local conservation properties of multi-symplectic spectral discretizations.