SciPost logo

A new integrable structure associated to the Camassa-Holm peakons

Jean Avan, Luc Frappat, Eric Ragoucy

SciPost Phys. 15, 228 (2023) · published 6 December 2023


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.

Cited by 1

Crossref Cited-by

Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.