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 bounda ry 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 ca se, a direct perturbation approach for dark solitons is developed in t his paper. Difficulties caused by the background are overcome. As an e xample of the approach, the problem of damping is treated as a perturb ation.