GENERAL SOLITON-SOLUTIONS OF THE NLS-CONDITION( EQUATION UNDER NONVANISHING BOUNDARY)

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 systematic ally 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 p hase shift of the peak are derived. When the boundary value approaches zero, all the formulas become the known ones naturally.