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A new integrable structure associated to the Camassa-Holm peakons

by J. Avan, L. Frappat, E. Ragoucy

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Eric Ragoucy
Submission information
Preprint Link: scipost_202310_00041v2  (pdf)
Date accepted: 2023-11-27
Date submitted: 2023-11-16 18:42
Submitted by: Ragoucy, Eric
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


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.

Author comments upon resubmission

We corrected the misprints pointed out by the referee

List of changes

The essential part is the correction of the r-matrix in eq. 4.22

Published as SciPost Phys. 15, 228 (2023)

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