On Calogero-Francoise-type Lax matrices and their dynamical r-matrices

TL;DR

This paper introduces new integrable systems related to Calogero-Francoise flows, featuring dynamical r-matrices that depend on canonical variables, expanding the understanding of peakon-type models.

Contribution

It presents a novel class of classical integrable systems of Camassa-Holm peakon type with explicit dynamical r-matrices, generalizing previous Calogero-Francoise flows.

Findings

Derived new integrable peakon systems with periodic potentials

Computed explicit dynamical r-matrices depending on canonical variables

Extended the class of known integrable models with maximal even D_2 symmetry

Abstract

New classical integrable systems of Camassa-Holm peakon type are proposed. They realize the maximal even piecewise-D_2 generalization of the Calogero-Francoise flows, yielding periodic and pseudoperiodic trigonometric/hyperbolic potentials. The associated r-matrices are computed. They are dynamical and depend on both sets {p_i} and {q_i} of canonical variables.