A new physically meaningful general nonlocal reverse-space nonlinear Schrödinger equation and its novel Riemann-Hilbert method via temporal-part spectral analysis for deriving soliton solutions

By imposing a nonlocal reverse-space symmetry constraint on a general coupled nonlinear Schrodinger (NLS) equation, we propose a new general nonlocal reverse-space NLS equation with two free real parameters involving the effects of the self-phase modulation, the cross-phase modulation and the four-wave mixing. The proposed nonlocal equation is physically meaningful in two aspects. One is that, by solving the proposed nonlocal equation, one can obtain corresponding solutions of the general coupled NLS equation with special initial conditions. The other is that the proposed nonlocal equation is an integrable generalization of a physically significant nonlocal reverse-space NLS equation in the literature. For the proposed nonlocal equation, we develop a novel Riemann-Hilbert (RH) method where the spectral analysis is performed from the temporal part of the Lax pair rather than the spatial part as in the traditional RH approach. Firstly, the complicated spectral symmetry structure of the proposed nonlocal equation is explored in detail. Secondly, by solving the RH problem with the complicated spectral symmetry structure, soliton solutions are rigorously obtained for the nonlocal equation. Thirdly, some new soliton dynamical behaviors underlying the soliton solutions are theoretically investigated and graphically simulated.

Automated summary

A new general nonlocal reverse-space nonlinear Schrodinger equation is proposed by imposing a nonlocal reverse-space symmetry constraint on a general coupled NLS equation. The equation is physically meaningful and can be used to obtain corresponding solutions of the general coupled NLS equation with special initial conditions. A novel Riemann-Hilbert method is developed to solve the proposed nonlocal equation, and soliton solutions are rigorously obtained by solving the RH problem with the complicated spectral symmetry structure. Some new soliton dynamical behaviors underlying the soliton solutions are also theoretically investigated and graphically simulated.