Within the framework of the Riemann-Hilbert problem, the theory of inverse scattering transform is established for the defocusing nonlinear Schrödinger equation with local and nonlocal nonlinearities (which originates from the parity-symmetric reduction of the Manakov system) under non-zero boundary conditions. First, the adjoint Lax pair and auxiliary eigenfunctions are introduced for the direct scattering, and the analyticity, symmetries of eigenfunctions and scattering matrix are studied in detail. Then, the distribution of discrete eigenvalues is examined, and the asymptotic behaviors of the eigenfunctions and scattering coefficients are analyzed rigorously. Compared with the Manakov system, the reverse-space nonlocality introduces an additional symmetry, leading to stricter constraints on eigenfunctions, scattering coefficients and norming constants. Further, the Riemann-Hilbert problem is formulated for the the inverse problem with the scattering coefficients admitting an arbitrary number of simple zeros. For the reflectionless case, the N-soliton solutions are presented in the determinant form. With N=1, the dark and beating one-soliton solutions are obtained, which are respectively associated with a pair of discrete eigenvalues lying on and off the circle on the spectrum plane. Via the asymptotic analysis, the two-soliton solutions are found to admit the interactions between two dark solitons or two beating solitons, as well as the superpositions of two beating solitons or one beating soliton and one dark soliton.
Xu et al. (Sun,) studied this question.