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

@article{Sykes2011ExactST, title={Exact solutions to the four Goldstone modes around a dark soliton of the nonlinear Schr{\"o}dinger equation}, author={Andrew G. Sykes}, journal={Journal of Physics A}, year={2011}, volume={44}, pages={135206} }

This paper is concerned with the linearization around a dark soliton solution of the nonlinear Schr?dinger 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…

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