In this talk I will study the long time behavior of discrete
approximations in time and space of the cubic nonlinear Schroedinger
equation on the real line.
Considering a symplectic time-splitting approximation of discrete
nonlinear Schroedinger equations with Dirichlet boundary condition on
a large space interval, we can prove that under a CFL condition
between the time and space discretization parameters, there exists a
numerical solution which is close to the continuous ground state. Then
we can prove that if the initial datum is close to the ground state,
the associated numerical solution remains close to the orbits of the
ground state over a very long time. This is a joint work with Dario
Bambusi (Milano) and Benoit Grebert (Nantes)