Generalized Darboux Transformation and Analytic Solutions in an Inhomogeneous Variable-Coefficient Discrete Hirota Equation

This paper introduces and investigates a novel inhomogeneous variable-coefficient discrete Hirota equation which is a combination of discrete nonlinear Schrödinger equation and discrete complex modified Korteweg–de Vries equations. Through spectral analysis, the related compatibility conditions and Lax pairs are explicitly derived, on which basis we successfully construct a generalized (n,N−n)-fold discrete Darboux transformation. Leveraging this newly established gauge matrix framework, we obtain and graphically characterize several families of exact localized solutions, including non-trivial solitons and breathers. Our visual simulations reveal that the dynamic trajectories and structural profiles of these localized patterns are strongly governed by the external potential functions. Furthermore, by employing the algebraic infrastructure of the Tu scheme, we derive a novel integrable discrete Hirota hierarchy associated with the given spectral problem, subsequently establishing its rigid Hamiltonian formulation alongside an infinite set of conservation laws.