Nonlocal Reductions of The Multicomponent Nonlinear Schrödinger Equation on Symmetric Spaces

Abstract

Our aim is to develop the inverse scattering transform for multicomponent generalizations of nonlocal reductions of the nonlinear Schrödinger (NLS) equation with \(\mathcal{PT}\) symmetry related to symmetric spaces. This includes the spectral properties of the associated Lax operator, the Jost function, the scattering matrix, the minimum set of scattering data, and the fundamental analytic solutions. As main examples, we use theManakov vector Schrödinger equation (related to A.III-symmetric spaces) and the multicomponent NLS (MNLS) equations of Kulish–Sklyanin type (related to BD.I-symmetric spaces). Furthermore, we obtain one- and two-soliton solutions using an appropriate modification of the Zakharov–Shabat dressing method. We show that the MNLS equations of these types admit both regular and singular soliton configurations. Finally, we present different examples of one- and two-soliton solutions for both types of models, subject to different reductions.