标题： 半离散复化Korteweg-de Vries方程在零边界条件和非零边界条件下的黎曼-希尔伯特方法与N孤子解 显示英文标题

标题： The semi-discrete complex modified Korteweg-de Vries equation with zero and non-zero boundary conditions: Riemann-Hilbert approach and N-soliton solutions

摘要： 本文专注于半离散复改良Korteweg-de Vries (DcmKdV)方程。 直接和逆散射理论在势的零边界条件和非零边界条件 (BCs) 下被建立。 对于直接问题，研究了特征函数和散射矩阵的性质，包括解析性、渐近性和对称性，这有助于建立黎曼-希尔伯特 (RH) 问题。 通过求解逆问题部分的RH问题，得到重构势公式，这使我们能够在无反射情况下推导出N孤子解。 同时，通过研究相应的RH问题推导出迹公式。 此外，通过图形模拟展示了零边界和非零边界条件下1孤子和2孤子的动力特性。 显示英文摘要

摘要： We focus on the semi-discrete complex modified Korteweg-de Vries (DcmKdV) equation in this paper. The direct and inverse scattering theory is developed with zero and non-zero boundary conditions (BCs) of the potential. For direct problem, the properties of the eigenfunctions and the scattering matrix, including analyticity, asymptotics and symmetries, are investigated, which facilitates the establishment of the Riemann-Hilbert (RH) problems. By solving the RH problems in the inverse problem part, the reconstruction potential formulas are obtained, which allows us to derive the N-soliton solutions in the reflectionless case. Meanwhile, the trace formulas are derived by means of studying the corresponding RH problems. Furthermore, the dynamic characteristics of the 1-soliton and 2-soliton with zero and non-zero boundary are demonstrated by graphical simulation.