On the construction of almost periodic solutions for the derivative nonlinear Schr\"odinger equation

TL;DR

This paper proves that for most potentials, the derivative nonlinear Schrödinger equation on a torus admits solutions that are almost periodic in time, expanding understanding of its long-term behavior.

Contribution

It establishes the existence of almost-periodic solutions for the derivative nonlinear Schrödinger equation for generic potentials on the torus.

Findings

Almost all potentials admit almost-periodic solutions.

The solutions are constructed on the torus domain.

The results apply to a broad class of potentials.

Abstract

In this paper, we consider a derivative nonlinear Schr\"odinger equation $$ \mathrm{i}\partial_{t}u+\partial_{xx}u-V\ast u+\mathrm{i}\vert u\vert^{2}\partial_{x}u=0 $$ on the torus $\mathbb{T}$, depending on some potential $V$. We prove that for `almost all' potentials $V$, this equation admits an almost-periodic solution.