Solitary Waves of the Two-Dimensional Camassa-Holm--Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates solitary wave solutions in a (2+1)-dimensional Camassa-Holm nonlinear Schrödinger equation, deriving approximate solutions via KP equations and validating them through numerical simulations.

Contribution

It introduces a novel approach to approximate solitary waves in the CH-NLS model using multiscale expansion and KP solitons, including validation through numerical analysis.

Findings

Constructed dark and anti-dark solitary wave solutions.

Validated approximate solutions with numerical simulations.

Analyzed wave interactions and stability.

Abstract

In this work, we study solitary waves in a (2+1)-dimensional variant of the defocusing nonlinear Schr\"odinger (NLS) equation, the so-called Camassa-Holm NLS (CH-NLS) equation. We use asymptotic multiscale expansion methods to reduce this model to a Kadomtsev--Petviashvili (KP) equation. The KP model includes both the KP-I and KP-II versions, which possess line and lump soliton solutions. Using KP solitons, we construct approximate solitary wave solutions on top of the stable continuous-wave solution of the original CH-NLS model, which are found to be of both the dark and anti-dark type. We also use direct numerical simulations to investigate the validity of the approximate solutions, study their evolution, as well as their head-on collisions.