Infinite dimensional invariant tori for nonlinear Schr\"odinger equations

TL;DR

This paper demonstrates the existence of infinite-dimensional invariant tori for nonlinear Schrödinger equations on the circle, extending classical KAM results to an infinite-dimensional setting and addressing a longstanding open problem.

Contribution

It proves the existence of infinite-dimensional non-resonant invariant tori near finite-dimensional KAM tori for nonlinear Schrödinger equations, advancing the understanding of their long-term dynamics.

Findings

Existence of infinite-dimensional invariant tori near finite-dimensional KAM tori

Construction of non-resonant Kronecker tori in PDEs

Answers a longstanding open question in Hamiltonian PDE dynamics

Abstract

We prove that nonlinear Schr\"odinger equations on the circle, without external parameters, admits plenty of almost periodic solutions. Indeed, we prove that arbitrarily close to most of the finite dimensional KAM tori constructed by Kuksin--Poschel in 1996, there exist infinite dimensional non resonant Kronecker tori, i.e. rotational invariant tori. This result answers a natural and longstanding question, well identified by the Hamiltonian PDE community since the first KAM-type result for PDEs by Kuksin in 1987.