  In this paper we study the long time behavior of a discrete approximation in
time and space of the cubic nonlinear Schr\"odinger equation on the real line.
More precisely, we consider a symplectic time splitting integrator applied to a
discrete nonlinear Schr\"odinger equation with additional Dirichlet boundary
conditions on a large interval. We give conditions ensuring the existence of a
numerical soliton which is close in energy norm to the continuous soliton. Such
result is valid under a CFL condition between the time and space stepsizes.
Furthermore we prove that if the initial datum is symmetric and close to the
continuous soliton, then the associated numerical solution remains close to the
orbit of the continuous soliton for very long times.
