Dynamics of the black soliton in a regularized nonlinear Schrodinger equation

TL;DR

This paper investigates the stability of black solitons in a regularized nonlinear Schrödinger equation, establishing spectral stability thresholds and illustrating dynamics through numerical simulations.

Contribution

It provides the first explicit spectral stability criteria for black solitons in a regularized NLS and explores their dynamics via numerical methods.

Findings

Spectral stability depends on the regularization parameter.

Black solitons are stable below a certain threshold.

Numerical simulations illustrate stable and unstable behaviors.

Abstract

We consider a family of regularized defocusing nonlinear Schrodinger (NLS) equations proposed in the context of the cubic NLS equation with a bounded dispersion relation. The time evolution is well-posed if the black soliton is perturbed by a small perturbation in the Sobolev space $H^s(\R)$ with s > 1/2. We prove that the black soliton is spectrally stable (unstable) if the regularization parameter is below (above) some explicitly specified threshold. We illustrate the stable and unstable dynamics of the perturbed black solitons by using the numerical finite-difference method. The question of orbital stability of the black soliton is left open due to the mismatch of the function spaces for the energy and momentum conservation.