A Note on the Symplectic Integration of the Nonlinear Schrödinger Equation

Numerically solving the nonlinear Schrödinger equation and being able to treat arbitrary space dependent potentials permits many application in the realm of quantum mechanics. The long-term stability of a numerical method and its conservation properties is an important feature since it assures that the underlying physics of the solution are respected and it ensures that the numerical result is correct also for small time spans. In this paper we describe symplectic integrators for the nonlinear Schrödinger equation with arbitrary potentials and perform numerical experiments comparing different approaches and highlighting their respective advantages and disadvantages.