Spectral structures and soliton dynamical behaviors of two shifted nonlocal NLS equations via a novel Riemann–Hilbert approach: A reverse-time NLS equation and a reverse-spacetime NLS equation

By extending the traditional Riemann–Hilbert (RH) approach of soliton equations to a novel version, this paper is devoted to studying two shifted nonlocal NLS equations: a reverse-time NLS equation with a temporal parameter t0, and a reverse-spacetime NLS equation with the spatial and temporal parameters x0,t0. It is shown that the defocusing cases of the two shifted nonlocal NLS equations are equivalent to their focusing cases under linear transformations, only the focusing cases of the two shifted nonlocal NLS equations need to be treated in this work which differs from the classical local NLS equation whose focusing and defocusing cases should be treated separately. Firstly, the spectral analysis of the Lax pair of the two shifted nonlocal NLS equations are performed, from which spectral structures of the two shifted equations are revealed in detail, respectively. Secondly, based on the obtained symmetry relations of the scattering data, the soliton solutions of the two shifted nonlocal NLS equations are rigorously obtained by introducing a novel reduction-proof technique which is different but more direct than the traditional RH approach. It is demonstrated that, due to the existence of the shifted parameters, the spectral analysis for deriving the symmetry relations of the scattering data and thus the calculations of the soliton solutions for the two shifted nonlocal NLS equations involve more tedious and ingenious computations than those unshifted ones without shifted parameters. Finally, the dynamical behaviors underlying the obtained soliton solutions are theoretically explored and graphically illustrated by highlighting the roles that the shifted parameters play, which manifest the peculiar soliton characteristics that may have potential applications of the shifted nonlocal reverse-time and reverse-spacetime NLS equations.