Long time stability of small finite gap solutions of the cubic Nonlinear Schr\"odinger equation on $\mathbb T^2$

TL;DR

This paper proves long-time orbital stability of certain quasi-periodic solutions to the cubic defocusing nonlinear Schrödinger equation on a 2D torus, using a normal form approach and KAM theory.

Contribution

It establishes explicit conditions under which these solutions are stable for long times, extending the understanding of stability in nonlinear Schrödinger equations.

Findings

Most solutions satisfy the stability conditions.

The stability holds for finite but long times.

The proof combines normal form, KAM, and algebraic analysis techniques.

Abstract

In this paper we study long time stability of a class of nontrivial, quasi-periodic solutions depending on one spacial variable of the cubic defocusing non-linear Schr\"odinger equation on the two dimensional torus. We prove that these quasi-periodic solutions are orbitally stable for finite but long times, provided that their Fourier support and their frequency vector satisfy some complicated but explicit condition, which we show holds true for most solutions. The proof is based on a normal form result. More precisely we expand the Hamiltonian in a neighborhood of a quasi-periodic solution, we reduce its quadratic part to diagonal constant coefficients through a KAM scheme, and finally we remove its cubic terms with a step of nonlinear Birkhoff normal form. The main difficulty is to impose second and third order Melnikov conditions; this is done by combining the techniques of reduction in order of pseudo-differential operators with the algebraic analysis of resonant quadratic Hamiltonians.