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