Inverse scattering transform for the defocusing nonlinear Schrödinger equation with fully asymmetric non-zero boundary conditions

We formulate the inverse scattering transform (IST) for the defocusing nonlinear Schr¨odinger (NLS) equation with fully asymmetric non-zero boundary conditions (i.e., when the magnitudes of the limiting values of the solution at space infinities are not the same). The theory is formulated without making use of Riemann surfaces, and by dealing explicitly instead with the branched nature of the eigenvalues of the associated scattering problem. For the direct problem, we give explicit single-valued definitions of the Jost eigenfunctions and scattering coefficients over the whole complex plane, and we characterize their discontinuous behavior across the branch cut arising from the square root jump of the corresponding eigenvalues. We write the inverse problem as a discontinuous Riemann Hilbert Problem on an open contour, and we reduce the problem to a standard set of linear integral equations. We also give an expression for the trace formula and asymptotic phase difference. Finally, for comparison purposes, we also present the single-sheet, branch cut formulation of the inverse scattering transform for the initial value problem with symmetric non-zero boundary conditions, and we also briefly describe the formulation of the inverse scattering transform when different choices are made for the location of the branch cuts.