Infinite dimensional invariant tori for nonlinear Schrödinger equations
Joackim Bernier, Benoit Grébert, Tristan Robert
TL;DR
This work proves the existence of non resonant infinite dimensional Kronecker tori for the nonlinear Schrödinger equation on the circle without external parameters, yielding almost periodic solutions that are not quasi-periodic. The authors introduce a regularizing normal form to gain nonlinear smoothing (modulo a gauge) and then implement an iterative Pöschel–type scheme adding one mode at a time, combining a tailored Birkhoff step with a KAM theorem that uses internal parameters to control small divisors. The main contributions include a precise extended version of the invariant-torus result, a framework of analytic Hamiltonians in partial action-angle variables, and a robust scheme that culminates in infinite-dimensional invariant tori accumulating on the finite-dimensional tori of KP96. The approach leverages dispersive smoothing, Wick renormalization, and a double-nested iteration (KAM within Birkhoff steps) to navigate the small-divisor problem and frequency modulation, suggesting potential applicability to other 1D dispersive PDEs without external parameters. The results advance the understanding of almost periodic dynamics in non-integrable Hamiltonian PDEs and demonstrate that infinite-dimensional invariant structures can persist in a controlled, constructive manner without external frequency parameters.
Abstract
We prove that nonlinear Schrödinger 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.