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 Schrödinger (NLS) equation with fully asymmetric non-zero boundary conditions (i.e., when the limiting values of the solution at space infinities have different non-zero moduli). The theory is formulated without making use of Riemann surfaces, and instead by dealing explicitly 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 behavior of the corresponding eigenvalues. We pose the inverse problem as a Riemann–Hilbert Problem on an open contour, and we reduce the problem to a standard set of linear integral equations. Finally, for comparison purposes, we present the single-sheet, branch cut formulation of the inverse scattering transform for the initial value problem with symmetric (equimodular) non-zero boundary conditions, as well as for the initial value problem with one-sided non-zero boundary conditions, and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts.