Separation of Variables, Quasi-Trigonometric $r$-Matrices and Generalized Gaudin Models

TL;DR

This paper introduces new methods for separating variables in classical integrable systems associated with quasi-trigonometric $r$-matrices, expanding the understanding of Gaudin models and their spectral curves.

Contribution

It constructs two new families of separated variables for systems governed by non-skew-symmetric, non-dynamical $rak{gl}(2) imes rak{gl}(2)$ $r$-matrices, and analyzes their spectral curves.

Findings

New separated variables for classical Lax-integrable systems.

Most separation curves differ from the standard spectral curve.

Application to $N=2$ quasi-trigonometric Gaudin models in a magnetic field.

Abstract

We construct two new one-parametric families of separated variables for the classical Lax-integrable Hamiltonian systems governed by a one-parametric family of non-skew-symmetric, non-dynamical $\mathfrak{gl}(2)\otimes \mathfrak{gl}(2)$-valued quasi-trigonometric classical $r$-matrices. We show that for all but one classical $r$-matrices in the considered one-parametric families the corresponding curves of separation differ from the standard spectral curve of the initial Lax matrix. The proposed scheme is illustrated by an example of separation of variables for $N=2$ quasi-trigonometric Gaudin models in an external magnetic field.