The initial value problem for the focusing Manakov system with nonzero boundary conditions at infinity is solved by developing an appropriate inverse scattering transform. The analyticity properties of the Jost eigenfunctions is investigated, and precise conditions on the potential that guarantee such analyticity are provided. The analyticity properties of the scattering coefficients is also established rigorously, and auxiliary eigenfunctions needed to complete the bases of analytic eigenfunctions are derived. The behavior of the eigenfunctions and scattering coefficients at the branch points is discussed, as are the symmetries of the analytic eigenfunctions and scattering coeffiecients. These symmetries are used to obtain a rigorous characterization of the discrete spectrum and to rigorously derive the symmetries of the associated norming constants. The asymptotic behavior of the Jost eigenfunctions is derived systematically. A general formulation of the inverse scattering problem as a Riemann-Hilbert problem is presented. Explicit relations among all reflection coefficients are given, and all entries of the scattering matrix are determined in the case of reflectionless solutions. New soliton solutions are explicitly constructed and discussed. These solutions, which have no analogue in the scalar case, are comprised of dark-bright soliton pairs as in the defocusing case. Finally, a consistent framework is formulated for obtaining solutions corresponding to any number of simple zeros of the analytic scattering coefficients, leading to any combination of bright and dark-bright soliton solutions.
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This work was partly supported by the National Science Foundation through grant DMS-1009453.
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