Almost Periodic Solutions of The Cubic Defocusing Nonlinear Schr\"odinger Equation

TL;DR

This paper proves the existence and uniqueness of almost periodic solutions to the cubic defocusing nonlinear Schrödinger equation with small quasiperiodic initial data, using spectral theory methods.

Contribution

It introduces a novel spectral analysis approach to establish almost periodic solutions for NLS with quasiperiodic initial data, addressing challenges in the NLS hierarchy.

Findings

Existence of almost periodic solutions for small analytic quasiperiodic initial data.

Pure absolutely continuous spectrum for associated Dirac operators with small potentials.

Spectral gaps decay exponentially and satisfy homogeneity and Craig-type conditions.

Abstract

This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schr\"odinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency conditions, the Cauchy problem admits a solution that is almost periodic in both space and time, and that this solution is unique among solutions locally bounded in a suitable sense. The analysis combines direct and inverse spectral theory. In the inverse spectral theory part, we prove existence, almost periodicity, and uniqueness for solutions with initial data whose associated Dirac operator has purely a.c.\ spectrum that is not too thin. This resolves novel challenges presented by the NLS hierarchy, such as an additional degree of freedom and an additional commuting flow. In the direct spectral theory part, for Dirac operators with small analytic quasiperiodic potentials with Diophantine frequency conditions, we prove pure a.c.\ spectrum, exponentially decaying spectral gaps, and spectral thickness conditions (homogeneity and Craig-type conditions).