1+1 Gaudin Model

TL;DR

This paper extends the Gaudin models to 1+1 dimensions, introducing equations of motion and Lax pairs for the ${ m sl}(N)$ case, and explores their relation to integrable systems like Landau-Lifshitz and principal chiral models.

Contribution

It develops 1+1 field generalizations of rational and elliptic Gaudin models, providing new equations, Lax pairs, and Hamiltonian densities, and analyzes their connections to known integrable models.

Findings

Introduced 1+1 Gaudin models with spectral parameter on Riemann sphere and elliptic curve.

Derived equations of motion and Lax pairs for ${ m sl}(N)$ and ${ m sl}(2)$ cases.

Connected the 2-site model to the principal chiral model.

Abstract

We study 1+1 field-generalizations of the rational and elliptic Gaudin models. For ${\rm sl}(N)$ case we introduce equations of motion and L-A pair with spectral parameter on the Riemann sphere and elliptic curve. In ${\rm sl}(2)$ case we study the equations in detail and find the corresponding Hamiltonian densities. The $n$-site model describes $n$ interacting Landau-Lifshitz models of magnets. The interaction depends on position of the sites (marked points on the curve). We also analyze the 2-site case in its own right and describe its relation to the principal chiral model. We emphasize that 1+1 version impose a restriction on a choice of flows on the level of the corresponding 0+1 classical mechanics.