On the field analogue of elliptic spin Calogero-Moser model: Lax pair and equations of motion

TL;DR

This paper develops a Lax pair for a field analogue of the elliptic spin Calogero-Moser model, deriving equations of motion and demonstrating reduction to known models, thus advancing the understanding of integrable systems with spin variables.

Contribution

It introduces a Lax pair construction for the field analogue of the elliptic spin Calogero-Moser model, including conditions for reduction to spinless cases.

Findings

Lax pair satisfies Zakharov-Shabat equation with an unwanted term

Equations of motion are derived on the unreduced phase space

Reduction yields the known spinless Calogero-Moser model

Abstract

The Lax pair for the field analogue of the classical spin elliptic Calogero-Moser is proposed. Namely, using the previously known Lax matrix we suggest an ansatz for the accompany matrix. The presented construction is valid when the matrix of spin variables ${\mathcal S}\in{\rm Mat}(N,\mathbb C)$ satisfies the condition ${\mathcal S}^2=c_0{\mathcal S}$ with some constant $c_0\in\mathbb C$. It is proved that the Lax pair satisfies the Zakharov-Shabat equation with unwanted term, thus providing equations of motion on the unreduced phase space. The unwanted term vanishes after additional reduction. In the special case ${\rm rank}(\mathcal S)=1$ we show that the reduction provides the Lax pair of the spinless field Calogero-Moser model obtained earlier by Akhmetshin, Krichever and Volvovski.