Traveling periodic waves and breathers in the nonlocal derivative NLS equation

TL;DR

This paper investigates the stability and explicit solutions of traveling periodic waves and breathers in a nonlocal derivative nonlinear Schrödinger equation, relevant for fluid dynamics and particle systems, providing new analytical characterizations and stability results.

Contribution

It introduces a comprehensive analysis of stability and explicit breather solutions for the nonlocal derivative NLS equation, including both focusing and defocusing cases, using Hirota's bilinear method.

Findings

Proves linear and nonlinear stability of the background in defocusing case.

Characterizes traveling periodic wave solutions explicitly.

Constructs breather solutions describing solitary waves on the background.

Abstract

A nonlocal derivative NLS (nonlinear Schr\"{o}dinger) equation describes modulations of waves in a stratified fluid and a continuous limit of the Calogero--Moser--Sutherland system of particles. For the defocusing version of this equation, we prove the linear stability of the nonzero constant background for decaying and periodic perturbations and the nonlinear stability for periodic perturbations. For the focusing version of this equation, we prove linear and nonlinear stability of the nonzero constant background under some restrictions. For both versions, we characterize the traveling periodic wave solutions by using Hirota's bilinear method, both on the nonzero and zero backgrounds. For each family of traveling periodic waves, we construct families of breathers which describe solitary waves moving across the stable background. A general breather solution with $N$ solitary waves propagating on the traveling periodic wave background is derived in a closed determinant form.