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.

7 pages, minor corrections

References in corpus (8)

• Relativistic Classical Integrable Tops and Quantum R-matrices
• Classical integrable systems and soliton equations related to eleven-vertex R-matrix
• Field analogue of the Ruijsenaars-Schneider model
• Higher rank 1+1 integrable Landau-Lifshitz field theories from associative Yang-Baxter equation
• On Elliptic Lax Systems on the Lattice and a Compound Theorem for Hyperdeterminants
• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve
• Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model
• Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation

Cited by in corpus (2)

• Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model
• Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain