On almost periodic solutions to NLS without external parameters
Joackim Bernier, Benoît Grébert
TL;DR
The paper addresses the existence of invariant, non-resonant, infinite-dimensional tori and almost periodic solutions for the nonlinear Schrödinger equation on the circle without external parameters. It develops a regularizing normal form that converts the nonlinear part into a smoothing perturbation (up to a gauge) and proves a convergent normalization by selectively removing large divisors, aided by Wick renormalization to handle resonant terms. Building on this, it implements an internal-parameter KAM scheme that iteratively adds one site to finite-dimensional tori, producing a continuum of infinite-dimensional Kronecker tori that accumulate on KP96’s finite-dimensional tori; this yields almost periodic solutions that are not quasi-periodic. The approach combines a δ-regularized smoothing mechanism with a carefully controlled twist condition and a sparsity constraint on the limiting index set, enabling a parameter-free, infinite-dimensional KAM construction with no external potentials. This extends KAM-type results for PDEs to external-parameter-free settings and demonstrates the existence of rich almost periodic dynamics for not-fully-integrable dispersive equations.
Abstract
In this note, we present a result established in [BGR24] where we prove that nonlinear Schrodinger equations on the circle, without external parameters, admit plenty of infinite dimensional non resonant invariant tori, or equivalently, plenty of almost periodic solutions. Our aim is to propose an extended sketch of the proof, emphasizing the new points which have enabled us to achieve this result.