A New Multi-Symplectic Integration Method for the Nonlinear Schrödinger Equation

We propose a new multi-symplectic integration method for the nonlinear Schrödinger equation. The new scheme is derived by concatenating spatial discretization of the multi-symplectic Fourier pseudospectral method with temporal discretization of a symplectic Euler scheme and it is semi-explicit in the sense that it does not need to solve the nonlinear algebraic equations at every time step. We verify that the multi-symplectic semi-discretization of the Schrödinger equation with periodic boundary conditions has N semi-discrete multi-symplectic conservation laws. The discretization in time of the semi-discretization leads to N full-discrete multi-symplectic conservation laws. Numerical results are presented to demonstrate the robustness and the stability.