Exact solutions to the four Goldstone modes around a dark soliton of the nonlinear Schroedinger equation

TL;DR

This paper derives exact analytical expressions for the four Goldstone modes around a dark soliton in the nonlinear Schrödinger equation, enabling precise analysis of perturbations and corrections in Bose-Einstein condensates.

Contribution

It provides the first explicit analytic solutions for the four zero eigenvalue modes around a dark soliton, and constructs a Greens matrix for response analysis.

Findings

Derived explicit Goldstone modes for dark solitons

Constructed Greens matrix for linear response

Applied to BEC wavefunction corrections

Abstract

This article is concerned with the linearisation around a dark soliton solution of the nonlinear Schr\"odinger equation. Crucially, we present analytic expressions for the four linearly-independent zero eigenvalue solutions (also known as Goldstone modes) to the linearised problem. These solutions are then used to construct a Greens matrix which gives the first-order spatial response due to some perturbation. Finally we apply this Greens matrix to find the correction to the dark-soliton wavefunction of a Bose-Einstein condensate in the presence of fluctuations.