This article is cited in 11 papers

Nonlocal reductions of the multicomponent nonlinear Schrödinger equation on symmetric spaces

G. G. Grahovski, A. J. Mustafa, H. Susanto

Department of Mathematical Sciences, University of Essex, Colchester, UK

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 the Manakov vector Schrödinger equation (related to A.III-symmetric spaces) and the multicomponent NLS (MNLS) equations of Kullish–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.

Keywords: integrable system, multicomponent nonlinear Schrödinger equation, Lax representation, Zakharov–Shabat system, spectral decompositions, $\mathcal{PT}$ symmetry, inverse scattering transform, Riemann–Hilbert problem, dressing method.

Received: 08.11.2017

DOI: 10.4213/tmf9507

 English version: Theoretical and Mathematical Physics, 2018, 197:1, 1430–1450

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