Perturbations of Dark Solitons

TL;DR

This paper introduces a method to approximate dark soliton solutions of the nonlinear Schrödinger equation under perturbations, analyzing the development of a propagating shelf around the soliton in various background conditions.

Contribution

It presents a novel approximation technique that separates the problem into inner and outer regions, incorporating conservation laws to analyze perturbations affecting dark solitons.

Findings

Shelf develops around the soliton and propagates with background-dependent speed.

The method effectively models the impact of linear and nonlinear damping perturbations.

Applicable to both constant and slowly evolving backgrounds.

Abstract

A method for approximating dark soliton solutions of the nonlinear Schrodinger equation under the influence of perturbations is presented. The problem is broken into an inner region, where core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton which propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the nonlinear Schrodinger equation are used to approximate the shape of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated including linear and nonlinear damping type perturbations.