## Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation

Jing-Bo Chen and Meng-Zhao Qin

### 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 Schrödinger equation (NLS) on real space. We show that the multi-symplectic semi-discretization of the nonlinear Schrödinger 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.

Full Text (PDF) [161 KB]

### Key words

Multi-symplectic, Fourier pseudospectral method, nonlinear Schrödinger equation.

65M99.

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