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

Abstract: We study the non relativistic limit of the solutions of the cubic nonlinear Klein--Gordon (KG) equation on an interval with Dirichlet boundary conditions. We construct a family of 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 prove the convergence of KG solutions to NLS solutions only on compact time intervals.