The Calogero-Fran\c{c}oise integrable system: algebraic geometry, Higgs fields, and the inverse problem

TL;DR

This paper explores the Calogero-Fran ext{c}oise integrable system through algebraic geometry and Higgs bundles, providing solutions to the inverse problem and insights into the system's spectral properties.

Contribution

It introduces a Higgs bundle framework for the Calogero-Fran ext{c}oise system and solves the inverse problem using continued fractions, advancing understanding of its integrability.

Findings

Solutions expressed as Higgs bundles over the projective line

Established linearization of isospectral flow

Connected the theta divisor to particle collisions in two-particle case

Abstract

We review the Calogero-Fran\c{c}oise integrable system, which is a generalization of the Camassa-Holm system. We express solutions as (twisted) Higgs bundles, in the sense of Hitchin, over the projective line. We use this point of view to (a) establish a general answer to the question of linearization of isospectral flow and (b) demonstrate, in the case of two particles, the dynamical meaning of the theta divisor of the spectral curve in terms of mechanical collisions. Lastly, we outline the solution to the inverse problem for CF flows using Stieltjes' continued fractions.