Almost-periodic solutions to the NLS equation with smooth convolution potentials

TL;DR

This paper proves the existence of almost-periodic solutions for the one-dimensional NLS equation with smooth convolution potentials, extending previous results to potentials of arbitrary high regularity without smoothing the nonlinearity.

Contribution

It establishes the existence of almost-periodic solutions for NLS with highly regular convolution potentials, a novel result in the field.

Findings

Existence of almost-periodic solutions in Gevrey class

Applicable to convolution potentials of any polynomial regularity

Solutions satisfy a Bryuno non-resonance condition

Abstract

We consider the one-dimensional NLS equation with a convolution potential and a quintic nonlinearity. We prove that, for most choices of potentials with polynomially decreasing Fourier coefficients, there exist almost-periodic solutions in the Gevrey class with frequency satisfying a Bryuno non-resonance condition. This allows convolution potentials of class $C^p$, for any integer $p$: as far as we know this is the first result where the regularity of the potential is arbitrarily large and not compensated by a corresponding smoothing of the nonlinearity.