Exact Bogoliubov-de Gennes Solutions for Grey Soliton Backgrounds

TL;DR

This paper derives exact solutions to the Bogoliubov-de Gennes equations for small excitations around grey solitons in the nonlinear Schrödinger equation, extending previous work on dark solitons.

Contribution

It provides a complete set of solutions for grey soliton backgrounds, including zero and non-zero frequency modes, using a novel factorization approach.

Findings

Derived exact solutions for grey soliton excitations

Extended solutions from dark to grey soliton backgrounds

Analyzed zero mode and non-zero frequency solutions

Abstract

We derive and discuss the complete set of exact solutions to the one-dimensional Bogoliubov-de Gennes equations for small amplitude excitations around general grey soliton solutions to the cubic nonlinear Schr\"odinger equation. Our results extend the previously known case of the motionless dark soliton background. We derive our non-zero frequency solutions using a variant of the factorization method for Schr\"odinger equations with reflectionless potentials. We also discuss the zero mode solutions at length.