We consider solutions of the matrix Kadomtsev–Petviashvili (KP) hierarchy that are elliptic functions of the first hierarchical time t₁ = x. It is known that poles x_i and matrix residues at the poles ${\rho }_i^{\alpha \beta }=a_i^{\alpha }{b}_i^{\beta }$ of such solutions as functions of the time t₂ move as particles of spin generalization of the elliptic Calogero–Moser model (elliptic Gibbons–Hermsen model). In this paper, we establish the correspondence with the spin elliptic Calogero–Moser model for the whole matrix KP hierarchy. Namely, we show that the dynamics of poles and matrix residues of the solutions with respect to the kth hierarchical time of the matrix KP hierarchy is Hamiltonian with the Hamiltonian H_k obtained via an expansion of the spectral curve near the marked points. The Hamiltonians are identified with the Hamiltonians of the elliptic spin Calogero–Moser system with coordinates x_i and spin degrees of freedom $a_i^{\alpha },b_i^{\beta }$.

中文翻译：

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矩阵 KP 层次的椭圆解和椭圆 Calogero-Moser 模型的自旋泛化

我们考虑矩阵 Kadomtsev-Petviashvili (KP) 层次的解，它们是第一层次时间t ₁ = x 的椭圆函数。已知极点x _i和极点处的矩阵残差${\rho }_{\mathrm{一世}}^{\alpha \beta }={\mathrm{一种}}_{\mathrm{一世}}^{\alpha }{乙}_{\mathrm{一世}}^{\beta }$作为时间t _{2 的}函数的此类解的t ₂作为椭圆 Calogero-Moser 模型（椭圆 Gibbons-Hermsen 模型）的自旋泛化粒子移动。在本文中，我们建立了与整个矩阵 KP 层次结构的自旋椭圆 Calogero-Moser 模型的对应关系。也就是说，我们证明了关于矩阵 KP 层次结构的第k层级时间的解的极点和矩阵残差的动力学是哈密顿量，其中哈密顿量H _k通过扩展标记点附近的谱曲线获得。哈密​​顿量与坐标为x _i和自旋自由度的椭圆自旋 Calogero-Moser 系统的哈密顿量相同${\mathrm{一种}}_{\mathrm{一世}}^{\alpha },{乙}_{\mathrm{一世}}^{\beta }$.