Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation

Abstract

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schrödinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schrödinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state. Such result is valid under a CFL condition of the form $\tau h^{-2}\le C$ where $\tau$ and $h$ denote the time and space step size respectively. Furthermore we prove that if the initial datum is symmetric and close to the continuous ground state $\eta$ then the associated numerical solution remains close to the orbit of $\eta ,\Gamma =\cup _\alpha \{e^{i\alpha }\eta \}$, for very long times.