A new integrable structure associated to the Camassa-Holm peakons

J. Avan, L. Frappat, E. Ragoucy
Nonlinear Sciences, Exactly Solvable and Integrable Systems, Exactly Solvable and Integrable Systems (nlin.SI), Mathematical Physics (math-ph)
2023-09-14 16:00:00
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.
PDF: A new integrable structure associated to the Camassa-Holm peakons.pdf
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