Reducibility and nonlinear stability for a quasi-periodically forced NLS

TL;DR

This paper investigates the long-term stability of KAM tori in a quasi-periodically forced nonlinear Schrödinger equation on a 2D torus, providing reducibility results and precise frequency asymptotics to ensure stability.

Contribution

It introduces a novel reducibility approach and detailed frequency asymptotics for the forced NLS, enabling the verification of Melnikov conditions at any order.

Findings

Proved reducibility of the quasi-periodically forced NLS.

Established long-time stability of the origin.

Derived precise asymptotic expansions of frequencies.

Abstract

Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schr\"odinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2\pi \mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.