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arXiv · 1910.00434

Matrix KP hierarchy and spin generalization of trigonometric Calogero-Moser hierarchy

V. Prokofev·A. Zabrodin
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Abstract

We consider solutions of the matrix KP hierarchy that are trigonometric functions of the first hierarchical time $t_1=x$ and establish the correspondence with the spin generalization of the trigonometric Calogero-Moser system on the level of hierarchies. Namely, the evolution of poles $x_i$ and matrix residues at the poles $a_i^{\alpha}b_i^{\beta}$ of the solutions with respect to the $k$-th hierarchical time of the matrix KP hierarchy is shown to be given by the Hamiltonian flow with the Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the spin trigonometric Calogero-Moser system with coordinates $x_i$ and with spin degrees of freedom $a_i^{\alpha}, \, b_i^{\beta}$. By considering evolution of poles according to the discrete time matrix KP hierarchy we also introduce the integrable discrete time version of the trigonometric spin Calogero-Moser system.

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V. Prokofev, A. Zabrodin. 2019-09-29. Matrix KP hierarchy and spin generalization of trigonometric Calogero-Moser hierarchy. https://arxiv.org/abs/1910.00434

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