On the completely integrable Calogero type discretizations of nonlinear Lax integrable dynamical systems and the related Markov type co-adjoint orbits

TL;DR

This paper develops Calogero-type discretizations for nonlinear integrable systems, revealing their Lie-algebraic structures, conservation laws, and Poisson structures, thus advancing the understanding of integrable discretizations and their algebraic foundations.

Contribution

It introduces a Calogero matrix discretization scheme for integrable systems, connecting them with co-adjoint orbits on Markov co-algebras and deriving their conservation laws and Poisson structures.

Findings

Discretizations preserve integrability and Lie-algebraic structures.

Conservation laws and Poisson structures are derived from the approach.

Limiting procedures connect discrete systems to continuous functional spaces.

Abstract

The Calogero type matrix discretization scheme is applied to constructing the Lax type integrable discretizations of one wide enough class of nonlinear integrable dynamical systems on functional manifolds. Their Lie-algebraic structure and complete integrability related with co-adjoint orbits on the Markov co-algebras is discussed. It is shown that a set of conservation laws and the associated Poisson structure ensue as a byproduct of the approach devised. Based on the Lie algebras quasi-representation property the limiting procedure of finding the nonlinear dynamical systems on the corresponding functional spaces is demonstrated.