An algebraic approach to discrete time integrability

TL;DR

This paper introduces an algebraic framework for constructing fully discrete integrable systems on 2D lattices, deriving new models like discrete nonlinear Schrödinger and Ablowitz-Ladik, and exploring their quantum counterparts.

Contribution

It presents a novel algebraic method for systematically deriving discrete integrable systems using r-matrices and classical algebras, including quantum extensions.

Findings

Derived two fully discrete nonlinear Schrödinger type systems

Applied Darboux-dressing method for solution construction

Explored quantization of the discrete systems

Abstract

We propose the systematic construction of classical and quantum two dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrodinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz-Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two dimensional quantum lattice is thus also examined.