KAM for beating solutions of the quintic NLS

TL;DR

This paper proves the existence of quasi-periodic solutions for the quintic nonlinear Schrödinger equation on the circle, demonstrating nonlinear energy exchanges near resonant solutions using KAM theory.

Contribution

It introduces a KAM-based method to establish quasi-periodic solutions bifurcating from resonant states in the quintic NLS, extending previous normal form analyses.

Findings

Existence of quasi-periodic solutions near resonant states

Recurrent energy exchange between Fourier modes

Application of KAM theory to nonlinear PDEs

Abstract

We consider the nonlinear Schr\"{o}dinger equation of degree five on the circle $\mathbb{S}^1 = \mathbb{R}/2\pi$. We prove the existence of quasi-periodic solutions which bifurcate from "resonant" solutions (studied in [14]) of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect.