A novel Riemann–Hilbert formulation-based reduction method to an integrable reverse-space nonlocal Manakov equation and its applications

Abstract

In this paper, a novel Riemann–Hilbert (RH) formulation-based reduction method is developed for an integrable reverse-space nonlocal Manakov equation. Firstly, the scattering-data constraints of the reverse-space nonlocal Manakov equation are shown to be difficult to determine via the traditional RH method. Secondly, to obtain the scattering-data constraints of the reverse-space nonlocal Manakov equation, the traditional RH method is extended to an improved version which we call a novel RH formulation-based reduction method. Specifically, utilizing the RH formulation-based reduction method, the scattering-data constraints of the reverse-space nonlocal Manakov equation are determined to guarantee the required nonlocal symmetry reduction of the two-component Ablowitz–Kaup–Newell–Segur (AKNS) system. Moreover, the scattering-data constraints of the reverse-space nonlocal Manakov equation are compared with those of the Manakov equation. Thirdly, $N$-soliton solutions of the reverse-space nonlocal Manakov equation are obtained by imposing the obtained scattering-data constraints in those of the two-component AKNS system. Furthermore, the applications of our novel RH formulation-based reduction method are confirmed by applying it to another integrable nonlocal Manakov equation of reverse-spacetime type. Moreover, the scattering-data constraints of the reverse-spacetime nonlocal Manakov equation are further compared with those of the reverse-space nonlocal Manakov equation and the Manakov equation, respectively. Additionally, the nonlinear soliton features of the reverse-space nonlocal Manakov equation and the reverse-spacetime nonlocal Manakov equation are analyzed and classified in detail, respectively, according to different spectral parameter selections.