Time discretization of the spin Calogero-Moser model and the semi-discrete matrix KP hierarchy

Abstract

We introduce the discrete time version of the spin Calogero-Moser system. The equations of motion follow from the dynamics of poles of rational solutions to the matrix KP hierarchy with discrete time. The dynamics of poles is derived using the auxiliary linear problem for the discrete flow.

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

“…Multiply the first equation by b α i (p) and sum over α, then multiply the second equation by a α i (p), sum over α and take into account the constraint b ν i a ν i = 1. Subtracting the resulting equations, we eliminate ẋi (p) and obtain the equations of motion (1.4): These equations of motion generalize the ones for the rational Gibbons-Hermsen model obtained in [19]. They look like the Bethe ansatz equations for the quantum trigonometric Gaudin model "dressed" by the spin variables.…”

“…The limit µ → ∞ is the continuous limit. One can show, using the explicit expressions of the wave functions through the tau-function and some corollaries of the bilinear relation (see [19]) that the corresponding linear problems have the form…”

“…A simple calculation, similar to the one done in [19], shows that the compatibility condition of equations (3.6), (6.6) is the discrete Lax equation…”

“…Another result of this paper is the time discretization of the trigonometric spin Calogero-Moser (Gibbons-Hermsen) model. (The time discretization of the rational spin Calogero-Moser system within the same approach was suggested in [19].) Because of the precise correspondence between the trigonometric solutions of the matrix KP hierarchy and the trigonometric spin Calogero-Moser hierarchy, the integrable time discretization of the Calogero-Moser system and its spin generalization can be obtained from dynamics of poles of trigonometric solutions to semi-discrete soliton equations.…”