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

TL;DR

This paper introduces a gradient-holonomic method for analyzing the integrability of discrete nonlinear dynamical systems, focusing on Lax type structures and asymptotic solutions, with detailed application to a discrete nonlinear Schrödinger system.

Contribution

It develops a novel gradient-holonomic approach for Lax integrability analysis of differential-discrete systems, providing new tools for studying their asymptotic solutions and integrability properties.

Findings

Established a gradient identity for Lax equations.

Proved integrability of a discrete nonlinear Schrödinger system.

Analyzed asymptotic solutions within the proposed framework.

Abstract

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.