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

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 $τ$-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.