A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions

Because of the essential difficulty caused by the non-vanishing boundary condition, a systematic perturbation approach for dark solitons has not yet been found. Based on a rigorous proof of the completeness of the squared Jost solution with explicit expressions for one-soliton case, a direct perturbation approach for dark solitons is developed in this paper. Difficulties caused by the background are overcome. As an example of the approach, the problem of damping is treated as a perturbation.