Stability in integrable nonlocal nonlinear equations

TL;DR

This paper investigates the stability of soliton solutions in various nonlocal integrable equations, revealing scenarios where solutions are stable, unstable, or change stability depending on parameters.

Contribution

It provides the first analysis of linear stability for solitons in nonlocal integrable systems like the nonlocal Hirota and Alice-Bob equations.

Findings

Identification of stable and unstable soliton solutions

Discovery of parameter-dependent stability changes

Extension of stability analysis to nonlocal integrable models

Abstract

Recently a variety of nonlocal integrable systems has been introduced that besides fields located at particlar space-time points simultaneously also contain fields that are located at different, but symmetrically related, points. Here we investigate different types of soliton solutions with regard to their stability against linear pertubations obtained for the nonlocal version of the Hirota/nonlinear Schr\"odinger equation and the so-called Alice and Bob versions of the Korteweg-de Vries and Bousinesq equations. We encounter different types of scenarios: Solition solutions that are linearly stable or unstable and also solutions that change their stability properties depending on the parameter regime they are in.