Asymptotic Expansions and Solitons of the Camassa-Holm Nonlinear Schrodinger Equation

TL;DR

This paper investigates the asymptotic behavior and soliton solutions of a deformed nonlinear Schrödinger equation called the CH-NLS, using multiscale expansions, approximate KdV solutions, and numerical simulations to analyze soliton interactions.

Contribution

It introduces a novel asymptotic reduction of the CH-NLS equation to KdV equations and constructs approximate dark and antidark soliton solutions with validation through numerical simulations.

Findings

Dark and antidark solitons can form on a continuous-wave background.

Small-amplitude solitons undergo quasi-elastic head-on collisions.

Asymptotic solutions are validated by numerical simulations.

Abstract

We study a deformation of the defocusing nonlinear Schr\"odinger (NLS) equation, the defocusing Camassa- Holm NLS, hereafter referred to as CH-NLS equation. We use asymptotic multiscale expansion methods to reduce this model to a Boussinesq-like equation, which is then subsequently approximated by two Korteweg-de Vries (KdV) equations for left- and right-traveling waves. We use the soliton solution of the KdV equation to construct approximate solutions of the CH-NLS system. It is shown that these solutions may have the form of either dark or antidark solitons, namely dips or humps on top of a stable continuous-wave background. We also use numerical simulations to investigate the validity of the asymptotic solutions, study their evolution, and their head-on collisions. It is shown that small-amplitude dark and antidark solitons undergo quasi-elastic collisions.