Quadratic Poisson brackets for the Camassa--Holm peakons

TL;DR

This paper introduces quadratic Poisson brackets for the generalized Camassa--Holm peakon model, establishing its bi-Hamiltonian structure and connecting it to known models through spectral parameter limits.

Contribution

It derives quadratic Poisson brackets for the generalized Camassa--Holm peakon model using a halved spectral parameter r-matrix, extending the linear structure and linking to known models.

Findings

Established quadratic Poisson brackets for the model.

Revealed bi-Hamiltonian structure combining linear and quadratic brackets.

Connected the structure to the classical Camassa--Holm and Ragnisco--Bruschi models.

Abstract

We establish quadratic Poisson brackets for the generalized Camassa--Holm peakon structure introduced in \cite{AFR23}. The calculation is based on the halving of the spectral parameter dependent $r$-matrix used to define the linear Poisson structure of this model. This quadratic structure, together with the linear one, establish the bi-Hamiltonian structure of the generalized Camassa--Holm peakon model. \\ When the deformation parameter tends to $\pm2$, the spectral parameter dependence drops out, and we recover the linear and quadratic Poisson structure of the Camassa--Holm peakon model. \\ When the spectral parameter tends to the fixed points of the involution defining the halving, we recover the Ragnisco--Bruschi deformation of the Camassa--Holm peakon model, thereby establishing its quadratic Poisson structure.