An algebraic approach to discrete time integrability

Anastasia Doikou,Iain Findlay

doi:10.1088/1751-8121/abd3d6

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 Schrödinger 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.

Highlights

The fundamental paradigm in the frame of classical integrable systems is the AKNS scheme [1]
Quantization in this context is quite natural as the classical r-matrix is replaced by a quantum Rmatrix that obeys the quantum YBE, and the classical Poisson algebra is replaced by a quantum algebra [20]
To illustrate the algebraic approach we present two distinct fully discrete versions of the non-linear Schrodinger system (NLS)-type hierarchy based on the existence of classical and quantum r-matrices and the underlying deformed algebras: 1) the fully discrete version of the system introduced in [40, 41], which is the more natural discretization of the NLS-type systems (AKNS scheme generally) from the algebraic point of view, and is associated to a rational r-matrix

Summary

Introduction

The fundamental paradigm in the frame of classical integrable systems is the AKNS scheme [1]. It is related to various physical problems such as arrays of coupled nonlinear wave-guides in nonlinear optics and quasi-particle motion on a dimer among others We stress that this is the first time to our knowledge that a systematic construction of fully discrete space-time integrable systems based on the existence of a classical r-matrix is achieved. This fundamental idea is naturally extended to the quantum case and the two dimensional quantum lattice can be constructed. After we provide the general algebraic set up for fully discrete integrable systems we examine two prototypical systems that are discretizations of the NLS-type scheme For both examples the time components of the Lax pairs are constructed as representations of the quadratic temporal algebras. The quantum discrete NLS model, associated to the Yangian R-matrix, as well as the quantum AblowitzLadik model (or q bosons), associated to a trigonometric R-matrix, are considered as our prototypical quantum systems

Discrete time Integrability: algebraic formulation

The two dimensional quantum lattice