Small amplitude weak Sobolev almost periodic solutions for the 1d NLS

TL;DR

This paper constructs weak, Sobolev-regular almost periodic solutions for the 1D nonlinear Schrödinger equation, addressing a major open problem in KAM theory for PDEs by extending solutions beyond classical regularity.

Contribution

It introduces the first existence results for weak, Sobolev-regular almost periodic solutions in non integrable PDEs within KAM theory.

Findings

Constructed Sobolev almost periodic solutions for 1D NLS.

Many solutions are weak, not classical.

First such results for non integrable PDEs in KAM theory.

Abstract

All the almost periodic solutions for non integrable PDEs found in the literature are very regular (at least $C^\infty$) and, hence, very close to quasi periodic ones. This fact is deeply exploited in the existing proofs. Proving the existence of almost periodic solutions with finite regularity is a main open problem in KAM theory for PDEs. Here we consider the one dimensional NLS with external parameters and construct almost periodic solutions which have only Sobolev regularity both in time and space. Moreover many of our solutions are so only in a weak sense. This is the first result on existence of weak, i.e. non classical, solutions for non integrable PDEs in KAM theory.