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

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 $\mathcal{S}\in \mathrm{Mat}(N,\mathbb{C})$ 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 $\mathrm{rank}(\mathcal{S})=1$ we show that the reduction provides the Lax pair of the spinless field Calogero-Moser model obtained earlier by Akhmetshin, Krichever and Volvovski.

17 pages, minor corrections

References in corpus (6)

• Field analogue of the Ruijsenaars-Schneider model
• Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations
• Higher rank 1+1 integrable Landau-Lifshitz field theories from associative Yang-Baxter equation
• Gauge equivalence between 1+1 rational Calogero-Moser field theory and higher rank Landau-Lifshitz equation
• Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model
• Gauge equivalence of 1+1 Calogero-Moser-Sutherland field theory and higher rank trigonometric Landau-Lifshitz model

Cited by in corpus (1)

• Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model