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A new integrable structure associated to the Camassa-Holm peakons
by J. Avan, L. Frappat, E. Ragoucy
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Submission summary
| Authors (as registered SciPost users): | Eric Ragoucy |
| Submission information | |
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| Preprint Link: | scipost_202310_00041v2 (pdf) |
| Date accepted: | Nov. 27, 2023 |
| Date submitted: | Nov. 16, 2023, 6:42 p.m. |
| Submitted by: | Ragoucy, Eric |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| 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
List of changes
The essential part is the correction of the r-matrix in eq. 4.22
Published as SciPost Phys. 15, 228 (2023)