Discrete nonlocal nonlinear Schrödinger systems: Integrability, inverse scattering and solitons

A number of integrable nonlocal discrete nonlinear Schrödinger (NLS) type systems have been recently proposed. They arise from integrable symmetry reductions of the well-known Ablowitz-Ladik scattering problem. The equations include: the classical integrable discrete NLS equation, integrable nonlocal: PT symmetric, reverse space time (RST), and the reverse time (RT) discrete NLS equations. Their mathematical structure is particularly rich. The inverse scattering transforms (IST) for the nonlocal discrete PT symmetric NLS corresponding to decaying boundary conditions was outlined earlier. In this paper, a detailed study of the IST applied to the PT symmetric, RST and RT integrable discrete NLS equations is carried out for rapidly decaying boundary conditions. This includes the direct and inverse scattering problem, symmetries of the eigenfunctions and scattering data. The general linearization method is based on a discrete nonlocal Riemann-Hilbert approach. For each discrete nonlocal NLS equation, an explicit one soliton solution is provided. Interestingly, certain one soliton solutions of the discrete PT symmetric NLS equation satisfy nonlocal discrete analogs of discrete elliptic function/Painlevé-type equations.