  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.
