Abstract
For the NLS+ equation under nonvanishing boundary condition, to avoid complexity due to Riemann surface, required for double-valued function of the usual spectral parameter, starting from a particular auxiliary spectral parameter, a special inverse scattering method is systematically developed, and explicit expressions of the general multi-soliton solutions are successfully obtained. Asymptotic behaviors of the multi-soliton solutions, additional displacement of center and additional phase shift of the peak are derived. When the boundary value approaches zero, all the formulas become the known ones naturally.
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