Riemann–Hilbert approach and nonlinear dynamics in the nonlocal defocusing nonlinear Schrödinger equation

Abstract

The nonlocal defocusing nonlinear Schrodinger (ND-NLS) equation is comparatively studied via the Riemann–Hilbert approach. Firstly, via spectral analysis, the spectral structure of the ND-NLS equation is investigated, which is different to those of the other three NLS-type equations, i.e., the local focusing nonlinear Schrodinger (LF-NLS) equation, the local defocusing nonlinear Schrodinger (LD-NLS) equation and the nonlocal focusing nonlinear Schrodinger (NF-NLS) equation. Secondly, by solving the Riemann–Hilbert problem corresponding to the reflectionless cases, multi-soliton solutions are obtained for the ND-NLS equation. Thirdly, we prove that, if parameters are suitably chosen, the multi-soliton solutions of the ND-NLS equation can be reduced to those of the LF-NLS equation and the LD-NLS equation, respectively. Fourthly, the multi-soliton solutions of the ND-NLS equation are demonstrated to possess repeated singularities generally, but they can also remain analytic for appropriate soliton parameters. Moreover, the multi-soliton dynamics are graphically illustrated using Mathematica symbolic computations. These results show that the solution structure and the nonlinear dynamics in the ND-NLS equation are rather different from those of the LF-NLS equation, the LD-NLS equation and the NF-NLS equation.