A number of integrable nonlocal discrete nonlinear Schrodinger (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/Painleve-type equations.

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离散非局部非线性薛定谔系统：可积性、逆散射和孤子

最近提出了许多可积分的非局部离散非线性薛定谔（NLS）型系统。它们源于著名的 Ablowitz-Ladik 散射问题的可积对称性减少。这些方程包括：经典的可积离散 NLS 方程、可积非局部：PT 对称、逆时空 (RST) 和逆时 (RT) 离散 NLS 方程。他们的数学结构特别丰富。对应于衰减边界条件的非局部离散 PT 对称 NLS 的逆散射变换 (IST) 已在前面概述。在本文中，针对快速衰减边界条件，对应用于 PT 对称、RST 和 RT 可积离散 NLS 方程的 IST 进行了详细研究。这包括直接和逆向散射问题，特征函数和散射数据的对称性。一般的线性化方法基于离散非局部 Riemann-Hilbert 方法。对于每个离散的非局部 NLS 方程，提供了一个显式的孤子解。有趣的是，离散 PT 对称 NLS 方程的某些孤子解满足离散椭圆函数/Painleve 型方程的非局部离散模拟。