Symplectic methods for the nonlinear Schrödinger equation

In this paper, we show that the spatial discretization of the nonlinear Schrödinger equation leads to a Hamiltonian system, which can be simulated with symplectic numerical schemes. In particular, we apply two symplectic integrators to the nonlinear Schrödinger equation, and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system, such as the energy and charge.