On the Complete Integrability of Nonlinear Dynamical Systems on Discrete Manifolds within the Gradient-Holonomic Approach

A gradient-holonomic approach for the Lax type integrability analysis of differentialdiscrete dynamical systems is devised. The asymptotical solutions to the related Lax equation are studied, the related gradient identity is stated. The integrability of a discrete nonlinear Schredinger type dynamical system is treated in detail.

💡 Research Summary

The paper introduces a novel “gradient‑holonomic” framework designed to analyze the Lax‑type integrability of differential‑discrete dynamical systems defined on discrete manifolds. Traditional Lax pair techniques, which are well‑established for continuous partial differential equations, encounter difficulties when directly applied to lattice‑type equations because of the coexistence of differential and difference operators. To overcome this, the authors combine variational calculus with holonomic (cohomological) concepts, constructing a unified formalism that simultaneously captures conserved quantities and underlying symmetries of the discrete system.

The core of the methodology is the definition of a Lax operator (L(\lambda)) and an auxiliary operator (M(\lambda)) on the lattice, where (\lambda) is a complex spectral parameter. The evolution equation (\dot L=