Higher rank 1+1 integrable Landau-Lifshitz field theories from associative Yang-Baxter equation

Abstract

We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\mathrm{gl}}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2=\mathrm{const}\cdot S$. The Lax pair with spectral parameter is constructed by means of a quantum $R$-matrix satisfying the associative Yang-Baxter equation. Equations of motion for ${\mathrm{gl}}_N$ Landau-Lifshitz model are derived from the Zakharov-Shabat equations. The model is simplified when $\mathrm{rank}(S)=1$. In this case the Hamiltonian description is suggested. The described family of models includes the elliptic model coming from ${\mathrm{GL}}_N$ Baxter-Belavin elliptic $R$-matrix. In $N=2$ case the widely known Sklyanin's elliptic Lax pair for XYZ Landau-Lifshitz equation is reproduced. Our construction is also valid for trigonometric and rational degenerations of the elliptic $R$-matrix.

8 pages, minor corrections

References in corpus (5)

• Relativistic Classical Integrable Tops and Quantum R-matrices
• Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations
• Classical integrable systems and soliton equations related to eleven-vertex R-matrix
• Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations
• Relativistic elliptic matrix tops and finite Fourier transformations

Cited by in corpus (8)

• 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
• Gauge equivalence between 1+1 rational Calogero-Moser field theory and higher rank Landau-Lifshitz equation
• On the field analogue of elliptic spin Calogero-Moser model: Lax pair and equations of motion
• Gauge equivalence of 1+1 Calogero-Moser-Sutherland field theory and higher rank trigonometric Landau-Lifshitz model
• Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model
• Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation
• Classical integrable spin chains of Landau-Lifshitz type from R-matrix identities