标题： 可积半离散非线性薛定谔方程与非零背景：双线化-约化方法 显示英文标题

标题： The integrable semi-discrete nonlinear Schrödinger equations with nonzero backgrounds: Bilinearization-reduction approach

摘要： 在本文中，通过双线性化-约简方法求解具有非零背景的经典和非局部半离散非线性薛定谔（sdNLS）方程。在该方法的第一步中，具有非零背景的未约简sdNLS系统被双线性化，并以准双卡萨蒂安的形式给出其解。然后，实施约简技术来处理复数和非局部约简，从而得到具有平面波背景或双曲函数背景的四个经典和非局部sdNLS方程的解。这些解用显式公式表示，并可根据某些谱矩阵的规范形式进行分类。特别是，我们给出了经典聚焦sdNLS方程的一般 rogue 波的显式公式。一些获得的解被分析并图示。 显示英文摘要

摘要： In this paper the classical and nonlocal semi-discrete nonlinear Schr\"{o}dinger (sdNLS) equations with nonzero backgrounds are solved by means of the bilinearization-reduction approach. In the first step of this approach, the unreduced sdNLS system with a nonzero background is bilinearized and its solutions are presented in terms of quasi double Casoratians. Then, reduction techniques are implemented to deal with complex and nonlocal reductions, which yields solutions for the four classical and nonlocal sdNLS equations with a plane wave background or a hyperbolic function background. These solutions are expressed with explicit formulae and allow classifications according to canonical forms of certain spectral matrix. In particular, we present explicit formulae for general rogue waves for the classical focusing sdNLS equation. Some obtained solutions are analyzed and illustrated.