Elliptic solutions to matrix KP hierarchy and spin generalization of elliptic Calogero-Moser model

Abstract

We consider solutions of the matrix KP hierarchy that are elliptic functions of the first hierarchical time $t_1=x$. 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 $t_2$ 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 $k$-th 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 $a_i^{\alpha },b_i^{\beta }$.

25 pages, no figures. arXiv admin note: text overlap with arXiv:1910.00434

References in corpus (4)

• Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy
• KP Trigonometric Solitons and an Adelic Flag Manifold
• Elliptic solutions to the KP hierarchy and elliptic Calogero-Moser model
• Matrix KP hierarchy and spin generalization of trigonometric Calogero-Moser hierarchy

Cited by in corpus (2)

• Integrable systems on multiplicative quiver varieties from cyclic quivers
• Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n{)}^{\ast }$-valued spins from Poisson reduction