A novel time-shifted nonlocal NLS equation with physical significance and its three types of multi-soliton solutions by Riemann–Hilbert approach

In this paper, we firstly propose a novel time-shifted nonlocal NLS equation with physical significance from a time-shifted nonlocal reduction of the Manakov system. Secondly, we devote our efforts to obtaining three types of multi-soliton solutions of the proposed equation by using the Riemann–Hilbert (RH) approach. Specifically, we establish the symmetry relations of the scattering data to characterize the spectral structure of the proposed equation. Then, three types of multi-soliton solutions are obtained, which correspond to three types of spectral zeros in the upper-half spectral plane: (i) all of the zeros are non-purely imaginary, (ii) all of the zeros are purely imaginary, (iii) some zeros are non-purely imaginary, some are purely imaginary. Thirdly, we make soliton analysis to theoretically reveal the soliton features underlying the obtained soliton solutions. Particularly, for the non-purely imaginary zeros, the two-soliton interactions are shown to be elastic. Moreover, the collision center of the two-soliton interaction is shown to be affected by both the spectral zero and the time-shifted parameter. For the purely imaginary zeros, the single-soliton solutions are shown to be stationary and they are not affected by the time-shifted parameter, while the two-solitons formulate periodic breathers. For the combination of non-purely imaginary and purely imaginary zeros, the three-soliton interactions exhibit polarization phenomena and the locations of soliton collisions are shown to be affected by the time-shifted parameter. Additionally, to illustrate the soliton features, we demonstrate the soliton solutions via choosing suitable spectral zeros and time-shifted parameter by highlighting the roles that these parameters play.