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

TL;DR

This paper investigates the non-relativistic limit of the cubic nonlinear Klein-Gordon equation, demonstrating that KAM solutions converge uniformly in time to solutions of the nonlinear Schrödinger equation, extending previous results to global time scales.

Contribution

It constructs a family of quasi-periodic solutions for the Klein-Gordon equation that converge globally in time to NLS solutions, using KAM theory, and extends uniform approximation results beyond compact time intervals.

Findings

Global uniform convergence of solutions in the non-relativistic limit

Construction of quasi-periodic solutions via KAM theory

Extension of approximation validity to all time

Abstract

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.