Scattering-data constraints, soliton solutions and dynamical behaviors of a shifted nonlocal Manakov equation by a novel improved Riemann–Hilbert approach

In this paper, an integrable variant of the Manakov equation, called the shifted nonlocal Manakov equation, is investigated by proposing a novel improved Riemann–Hilbert (RH) approach. Firstly, the scattering-data constraints of the shifted nonlocal Manakov equation are shown to be difficult to determine via the traditional RH approach, which is different from the Manakov equation whose scattering-data constraints are easy to obtain in terms of the RH approach. Secondly, to overcome the difficulties in deriving the scattering-data constraints of the shifted nonlocal Manakov equation, the traditional RH approach is extended to a novel version which we call a novel improved RH approach. Specifically, utilizing the novel improved RH approach, the scattering-data constraints of the shifted nonlocal Manakov equation are obtained to guarantee the required shifted nonlocal symmetry reduction of the two-component Ablowitz–Kaup–Newell–Segur (AKNS) system. Moreover, the scattering-data constraints of the shifted nonlocal Manakov equation are compared with those of the Manakov equation. Thirdly, N-soliton solutions of the shifted nonlocal Manakov equation are derived by imposing the scattering-data constraints in those of the two-component AKNS system. The merits of our novel improved RH approach lie in two aspects, (i) it can be applied to those nonlocal soliton equations whose scattering-data constraints are difficult to obtain via the traditional RH approach, (ii) it does not require the complicated spectral analysis involved in the traditional RH approach. In addition, it is theoretically proved that the obtained soliton solutions can produce both globally regular solitary behaviors and finite-time collapsing periodic behaviors depending on the parameter selections obeying the scattering-data constraints. Furthermore, the two-soliton interaction dynamical behaviors are also investigated which exhibit spatially localized and temporally periodic breather features. Finally, the soliton dynamical behaviors are illustrated with a few graphical simulations.