Classical Integrable Spin Chains of Landau–Lifshitz Type from R-Matrix Identities

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.