A note on stability of Eliasson-Kuksin's KAM tori for the nonlinear Schr\"{o}dinger equation

TL;DR

This paper improves Eliasson and Kuksin's KAM theorem for nonlinear Schrödinger equations on tori by using Kolmogorov's scheme, establishing local normal forms and demonstrating stability of invariant tori over time scales of order .

Contribution

It introduces a new approach using Kolmogorov's iterative scheme to enhance the stability analysis of KAM tori for nonlinear Schrödinger equations.

Findings

Established a local normal form for the Hamiltonian.

Proved -time stability of KAM tori.

Enhanced the KAM theorem with improved stability results.

Abstract

Eliasson and Kuksin developed a KAM approach to study the persistence of the invariant tori for nonlinear Schr\"{o}dinger equation on $\mathbb{T}^{d}$. In this note, we improve Eliasson and Kuksin's KAM theorem by using Kolmogorov's iterative scheme and obtain a local normal form for the transformed Hamiltonian. As a consequence, we are able to derive the time $\delta^{-1}$ stability of the obtained KAM tori.