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arXiv · 2309.08412

A new integrable structure associated to the Camassa-Holm peakons

J. Avan·L. Frappat·E. Ragoucy
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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.

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J. Avan, L. Frappat, E. Ragoucy. 2023-11-16. A new integrable structure associated to the Camassa-Holm peakons. https://doi.org/10.21468/scipostphys.15.6.228

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