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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
| Ontological classification |
| Academic field: |
Physics |
| Specialties:
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| Approach: |
Theoretical |
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.
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
