论文标题
较高的等级1+1来自联想Yang-baxter方程的可集成Landau-Lifshitz现场理论
Higher rank 1+1 integrable Landau-Lifshitz field theories from associative Yang-Baxter equation
论文作者
论文摘要
我们建议在$ {\ rm gl} _n $ case中建造1+1个可集成的海森伯格 - landau-lifshitz类型方程。动态变量是$ n \ times n $ matrix $ s $的矩阵元素，属性$ s^2 = {\ rm const} \ cdot s $。具有光谱参数的LAX对是通过满足Yang-Baxter方程的量子$ r $ -matrix来构建的。 $ {\ rm gl} _n $ landau-lifshitz模型的运动方程式来自Zakharov-Shabat方程。当$ {\ rm等级}（s）= 1 $时，该模型将简化。在这种情况下，提出了哈密顿的描述。所描述的模型系列包括来自$ {\ rm gl} _n $ baxter-belavin椭圆$ r $ -matrix的椭圆模型。在$ n = 2 $中，复制了XYZ Landau-Lifshitz方程的众所周知的Sklyanin的椭圆lax对。我们的构建也适用于椭圆形$ r $ -Matrix的三角和理性退化。
We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\rm gl}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2={\rm const}\cdot S$. 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 ${\rm gl}_N$ Landau-Lifshitz model are derived from the Zakharov-Shabat equations. The model is simplified when ${\rm rank}(S)=1$. In this case the Hamiltonian description is suggested. The described family of models includes the elliptic model coming from ${\rm GL}_N$ 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.
