Almost periodic invariant tori for the NLS on the circle

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.

arXiv comment44 pages

References in corpus (5)

• A lecture on the classical KAM theorem
• Long time stability of small finite gap solutions of the cubic Nonlinear Schrödinger equation on ${\mathbb{T}}^2$
• KAM for quasi-linear forced hamiltonian NLS
• KAM theory in configuration space and cancellations in the Lindstedt series
• The Stability of Full Dimensional KAM tori for Nonlinear Schrödinger equation

Cited by in corpus (2)

• On the construction of Sobolev Almost periodic invariant tori for the 1d NLS
• The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation