Stability of black solitons in optical systems with intensity-dependent dispersion

TL;DR

This paper investigates the stability of black solitons in optical systems with intensity-dependent dispersion, revealing unique spectral and energetic stability properties and their persistence under perturbations.

Contribution

It introduces new stability analysis results for black solitons in a modified NLS equation with intensity-dependent dispersion, including spectral and energetic stability conditions.

Findings

Spectral stability involves only isolated eigenvalues on the imaginary axis.

Black solitons remain stable under small decaying potentials.

Existence of traveling dark solitons for various wave speeds.

Abstract

Black solitons are identical in the nonlinear Schr\"{o}dinger (NLS) equation with intensity-dependent dispersion and the cubic defocusing NLS equation. We prove that the intensity-dependent dispersion introduces new properties in the stability analysis of the black soliton. First, the spectral stability problem possesses only isolated eigenvalues on the imaginary axis. Second, the energetic stability argument holds in Sobolev spaces with exponential weights. Third, the black soliton persists with respect to addition of a small decaying potential and remains spectrally stable when it is pinned to the minimum points of the effective potential. The same model exhibits a family of traveling dark solitons for every wave speed and we incorporate properties of these dark solitons for small wave speeds in the analysis of orbital stability of the black soliton.