Classical integrable spin chains of Landau-Lifshitz type from R-matrix identities

TL;DR

This paper introduces a new family of classical integrable spin chain models of Landau-Lifshitz type derived from R-matrix identities, connecting discrete and continuous integrable systems.

Contribution

It develops a novel ansatz for Lax pairs based on R-matrices satisfying the associative Yang-Baxter equation, leading to new integrable models.

Findings

Derived a family of space-discrete Landau-Lifshitz models

Connected discrete models to known continuous equations in the limit

Utilized R-matrix identities to obtain equations of motion

Abstract

We describe a family of 1+1 classical integrable space-discrete models of the Landau-Lifshitz type through the usage of ansatz for $U$-$V$ (Lax) pair with spectral parameter satisfying the semi-discrete Zakharov-Shabat equation. The ansatz for $U$-$V$ pair is based on $R$-matrices satisfying the associative Yang-Baxter equation and certain additional properties. Equations of motion are obtained using a set of $R$-matrix identities. In the continuous limit we reproduce the previously known family of the higher rank Landau-Lifshitz equations.