The foundation of the direct perturbation theory for solitons is a complete set of the squared Jost functions. With a suitable definition of the adjoint functions and inner products which yields orthogonal relations, the expansion of 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.
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Huang et al. (1999) studied this question.
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