Quasi-T\"oplitz functions in KAM theorem

TL;DR

This paper introduces Quasi-T"oplitz functions, proves an abstract KAM theorem for perturbations in this class, and applies it to the Non-Linear Schr"odinger equation on a torus to establish the existence and stability of quasi-periodic solutions.

Contribution

It defines Quasi-T"oplitz functions and proves a KAM theorem applicable to them, extending previous results to NLS equations with conserved momentum.

Findings

Existence of quasi-periodic solutions for NLS on torus

Stability results for these solutions

Simplified treatment using momentum conservation

Abstract

We define and describe the class of Quasi-T\"oplitz functions. We then prove an abstract KAM theorem where the perturbation is in this class. We apply this theorem to a Non-Linear-Scr\"odinger equation on the torus $T^d$, thus proving existence and stability of quasi-periodic solutions and recovering the results of [10]. With respect to that paper we consider only the NLS which preserves the total Momentum and exploit this conserved quantity in order to simplify our treatment.