A KAM algorithm for the resonant non--linear Schr\"odinger equation

TL;DR

This paper applies a KAM algorithm to the resonant nonlinear Schrödinger equation on a torus, constructing families of stable and unstable quasi-periodic solutions across multiple frequencies.

Contribution

It introduces a novel application of normal form techniques combined with KAM theory to generate quasi-periodic solutions for the resonant NLS in any dimension.

Findings

Construction of large families of quasi-periodic solutions.

Identification of both stable and unstable solutions.

Applicability to any number of independent frequencies.

Abstract

In this note we use the normal forms of the completely resonant non--linear Schr\"odinger equation on a torus (NLS) derived in previous work in order to produce, under a KAM algorithm, large families of stable and unstable quasi periodic solutions for the NLS in any number of independent frequencies.