Direct perturbation theory for dark solitons.

Perturbation theory for dark solitons governed by the nonlinear Schr\"odinger equation is developed. Orthogonality and closure of the basis of squared Jost functions are proved. The general form of a correction to the one-soliton solution up to first order is obtained. Two examples related to perturbed dynamics of dark optical solitons are considered within the framework of the adiabatic approximation.