Discrete Gerdjikov-Ivanov models and their higher-order counterparts from the Cauchy matrix scheme

Abstract

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.