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

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

2011-02-02

arxiv.org/abs/1102.0587v1

Physics

Condensed Matter

Quantum Gases

14 pages, 3 figures, submitted to Journal of Physics A

Scientific paper

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.

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