Quasi-Periodic solutions of Two Dimensional Completely Resonant Reversible Schr\"odinger d Systems

TL;DR

This paper develops an abstract KAM theorem for infinite-dimensional reversible Schrödinger systems and uses it with Birkhoff normal form to prove the existence of quasi-periodic solutions in resonant coupled nonlinear Schrödinger systems on a 2D torus.

Contribution

It introduces a novel KAM theorem tailored for infinite-dimensional reversible Schrödinger systems and demonstrates its application to find quasi-periodic solutions in resonant cases.

Findings

Existence of quasi-periodic solutions in a class of 2D coupled nonlinear Schrödinger systems.

Application of KAM and Birkhoff normal form methods to infinite-dimensional systems.

Establishment of conditions for resonance and reversibility in the solutions.

Abstract

We introduce an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible Schr\"odinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we find the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schr\"odinger d systems on two dimensional torus.