Non relativistic limit of the nonlinear Klein-Gordon equation: Uniform in time approximation of KAM solutions

We study the non relativistic limit of the solutions of the cubic nonlinear Klein--Gordon (KG) equation with periodic boundary conditions on an interval and we construct a family of time quasi periodic solutions which, after a Gauge transformation, converge globally uniformly in time to quasi periodic solutions of the cubic NLS. The proof is based on KAM theory. We emphasize that, regardless of the spatial domain, all the previous results concern approximations valid over compact time intervals.