Inverse Scattering Transform for the Vector NLS Equation with Non-vanishing Boundary Conditions

The inverse scattering transform for the vector defocusing vector nonlinear Schrodinger NLS equation with non-vanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. On the direct side, two out of the six scattering eigenfunctions do not admit an analytic extension on either sheet of the surface. Two additional analytic solutions are constructed by considering adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are discussed. In general a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions, which have dark solitonic behavior in both components and ones which have one dark and one bright component are constructed from the poles in the RH problem. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier solution obtained from the linearized vector NLS system.