AI 中文总结
本文在柯西矩阵框架下构建离散格尔季科夫 - 瓦诺夫模型及其高阶对应模型，通过特定方程推导主函数动力学并消除变量得多个模型族，给出显式解，经连续极限验证其与连续方程的关系，还研究了约化情况。
AI 中文摘要
格尔季科夫 - 伊万诺夫（GI）方程是导数非线性薛定谔系统中的重要模型，但其完全离散可积类似物尚未被探索。本文在柯西矩阵框架内系统地构建了GI方程及其高阶对应方程（hGI方程）的离散版本。从配备两组不同离散色散关系的西尔维斯特方程出发，推导主函数的移位动力学并消除辅助变量以获得封闭格点系统。由于消除步骤允许几个同样有效的代数恒等式，该过程产生了四个共轭对称的离散GI（dGI）模型族和四个离散高阶GI（dhGI）模型族。为每个离散模型分别通过具有对角和乔丹块谱矩阵的柯西矩阵方法提供了显式的N孤子和多极点解。通过两步连续极限验证，所有四个dGI模型都简化为同一个连续GI方程，所有四个dhGI模型都简化为同一个连续hGI方程。最后研究了约化：局部复共轭约化产生具有显式解的标量dGI和dhGI方程。此外，在高阶情况下，dhGI格点方程的成对重组允许非局部约化，产生非局部dhGI方程及其解。
英文摘要
The Gerdjikov-Ivanov (GI) equation is an important model in the derivative nonlinear Schrodinger system, yet its fully discrete integrable analogues remain unexplored. In this paper, we systematically construct discrete versions of both the GI equation and its higher-order counterpart (hGI equation) within the Cauchy matrix framework. Starting from the Sylvester equation equipped with two distinct sets of discrete dispersion relations, we derive the shift dynamics of the master functions and eliminate auxiliary variables to obtain closed lattice systems. Since the elimination step admits several equally valid algebraic identities, this procedure yields four conjugate-symmetric families of discrete GI (dGI) models and four families of discrete higher-order GI (dhGI) models. For each discrete model, we provide explicit N-soliton and multiple-pole solutions via the Cauchy matrix method with diagonal and Jordan-block spectral matrices, respectively. We verify through a two-step continuum limit, contracting one lattice direction at a time, that all four dGI models reduce to the same continuous GI equation and all four dhGI models reduce to the same continuous hGI equation. Finally, we investigate reductions: local complex conjugate reductions yield scalar dGI and dhGI equations with explicit solutions. Moreover, in the higher-order case, pairwise recombinations of the dhGI lattice equations admit nonlocal reductions that produce nonlocal dhGI equations and their solutions.
Comments33 pages
