A new integrable structure associated to the Camassa-Holm peakons

TL;DR

This paper introduces a new integrable Poisson algebra structure for peakon dynamics in the Camassa-Holm equation, leading to novel N-body solutions and explicit r-matrix formulations.

Contribution

It presents a new integrable Poisson algebra involving three matrices, extends it with a one-parameter family, and constructs explicit r-matrix formulations for peakon solutions.

Findings

New N-body peakon solutions depending on two parameters.

Explicit r-matrix formulations for the new Poisson algebra.

Identification of Poisson commuting quantities including original ones.

Abstract

We provide a closed Poisson algebra involving the Ragnisco--Bruschi generalization of peakon dynamics in the Camassa--Holm shallow-water equation. This algebra is generated by three independent matrices. From this presentation, we propose a one-parameter integrable extension of their structure. It leads to a new $N$-body peakon solution to the Camassa--Holm shallow-water equation depending on two parameters. We present two explicit constructions of a (non-dynamical) $r$-matrix formulation for this new Poisson algebra. The first one relies on a tensorization of the $N$-dimensional auxiliary space by a 4-dimensional space. We identify a family of Poisson commuting quantities in this framework, including the original ones. This leads us to constructing a second formulation identified as a spectral parameter representation.