Foundation of direct perturbation method for dark solitons

Abstract

The foundation of the direct perturbation theory for solitons is a complete set of the squared Jest functions. With a suitable definition of the adjoint functions and inner products which yields orthogonal relations, the expansion of the the unity is obtained and the completeness is shown by a generalized Marchenko equation. The direct perturbation method for dark solitons is generalized to the multi-soliton case.