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

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 TeX where TeX and TeX 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 TeX then the associated numerical solution remains close to the orbit of TeX , for very long times.