On nonlocal reductions of the multi-component nonlinear Schrodinger equation on symmetric spaces

TL;DR

This paper develops the inverse scattering transform for multi-component nonlocal NLS equations with PT-symmetry on symmetric spaces, providing spectral analysis, soliton solutions, and exploring regular and singular configurations.

Contribution

It introduces a comprehensive IST framework for multi-component nonlocal NLS equations on symmetric spaces, including soliton solutions and spectral properties, extending previous single-component approaches.

Findings

Derived spectral properties of the Lax operator.

Constructed 1- and 2-soliton solutions.

Identified regular and singular soliton configurations.

Abstract

The aim of this paper is to develop the inverse scattering transform (IST) for multi-component generalisations of nonlocal reductions of the nonlinear Schrodinger (NLS) equation with PT-symmetry related to symmetric spaces. This includes: the spectral properties of the associated Lax operator, Jost function, the scattering matrix and the minimal set of scattering data, the fundamental analytic solutions. As main examples, we use the Manakov vector Schr\"odinger equation (related to A.III-symmetric spaces) and the multi-component NLS (MNLS) equations of Kullish-Sklyanin type (related to BD.I-symmetric spaces). Furthermore, the 1- and 2-soliton solutions are obtained by using an appropriate modification of the Zakharov-Shabat dressing method. It is shown, that the MNLS equations of these types allow both regular and singular soliton configurations. Finally, we present here different examples of 1- and 2-soliton solutions for both types of models, subject to different reductions.