Almost periodic invariant tori for the NLS on the circle

TL;DR

This paper establishes the existence and stability of almost periodic solutions for the nonlinear Schrödinger equation on the circle, introducing a novel approach that broadens the class of solutions and PDEs analyzed.

Contribution

It presents a unified framework for proving the persistence of invariant tori in NLS, using a counter-term theorem directly in elliptic variables, extending previous results significantly.

Findings

Proves existence of almost periodic solutions for NLS on the circle.

Develops a new method avoiding action-angle variables.

Finds more solutions and applies to non-translation invariant PDEs.

Abstract

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain (2005) on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract "counter-term theorem" `a la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find "many more" almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.