The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces

TL;DR

This paper develops Hamiltonian representations for Lax-type hierarchies in differential-difference systems using Backlund transformations, analyzing their integrability, symmetries, and extensions.

Contribution

It introduces a Hamiltonian framework for Lax hierarchies on extended phase spaces and explores their relation to (2+1)-dimensional systems and linearizations.

Findings

Hamiltonian representation for Lax hierarchies established.

Relation between hierarchies and (2+1)-dimensional systems analyzed.

Existence of Hamiltonian structures on extended spaces demonstrated.

Abstract

The Hamiltonian representation for the hierarchy of Lax-type flows on a dual space to the Lie algebra of shift operators coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is found by means of a specially constructed Backlund transformation. The Hamiltonian description for the corresponding set of squared eigenfunction symmetry hierarchies is represented. The relation of these hierarchies with Lax integrable (2+1)-dimensional differential-difference systems and their triple Lax-type linearizations is analysed. The existence problem of a Hamiltonian representation for the coupled Lax-type hierarchy on a dual space to the central extension of the shift operator Lie algebra is solved also.