Zero curvature representation for classical lattice sine-Gordon equation via quantum R-matrix

TL;DR

This paper constructs local M-operators for the classical lattice sine-Gordon model using a quantum R-matrix, extending continuous-time representations to discrete space-time.

Contribution

It introduces a novel method to derive M-operators for the lattice sine-Gordon model via quantum R-matrix convolution, bridging quantum and classical integrable systems.

Findings

Construction of local M-operators from quantum R-matrix

Identification of vector components with tau-functions

Generalization of continuous-time representations to lattice models

Abstract

Local M-operators for the classical sine-Gordon model in discrete space-time are constructed by convolution of the quantum trigonometric 4$\times$4 R-matrix with certain vectors in its "quantum" space. Components of the vectors are identified with $\tau$-functions of the model. This construction generalizes the known representation of M-operators in continuous time models in terms of Lax operators and classical $r$-matrix.