Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 3

Special Issue in Honor of Claudio Procesi

Guest Editors: Luca Migliorini, Paolo Papi, and Mario Salvetti

Reducibility and nonlinear stability for a quasi-periodically forced NLS

Pages: 1313 – 1370

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n3.a8

Authors

E. Haus (Dipartimento di Matematica e Fisica, Università Roma Tre, Italy)

B. Langella (International School for Advanced Studies (SISSA), Trieste, Italy)

A. Maspero (International School for Advanced Studies (SISSA), Trieste, Italy)

M. Procesi (Dipartimento di Matematica e Fisica, Università Roma Tre, Italy)

Abstract

Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger 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.

Keywords

NLS equation, nonlinear stability

2010 Mathematics Subject Classification

35P15, 35J10, 37K55

Full Text (PDF format)

This research was supported by INDAM, PRIN 2020 (2020XB3EFL001) “Hamiltonian and dispersive PDEs”, PRIN 2022 (2022HSSYPN) “TESEO – Turbulent Effects vs Stability in Equations from Oceanography” and PRIN 2022 (F53D23002730006) “Stability in Hamiltonian dynamics and beyond”.

Received 1 August 2022

Accepted 27 December 2022

Published 15 May 2024