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VOLUME 115 (2022) | ISSUE 12 | PAGE 809
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  glN case. The dynamical variables are matrix elements of N× N matrix S with the property S2= const• 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  glN Landau-Lifshitz model are derived from the Zakharov-Shabat equations. The model is simplified when  rank(S)=1. In this case the Hamiltonian description is suggested. The described family of models includes the elliptic model coming from  GLN 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.