Quadratic Poisson brackets for the Camassa--Holm peakons
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912759654907904 |
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| author | Avan, J. Frappat, L. Ragoucy, E. |
| author_facet | Avan, J. Frappat, L. Ragoucy, E. |
| contents | We establish quadratic Poisson brackets for the generalized Camassa--Holm peakon structure introduced in \cite{AFR23}. The calculation is based on the halving of the spectral parameter dependent $r$-matrix used to define the linear Poisson structure of this model. This quadratic structure, together with the linear one, establish the bi-Hamiltonian structure of the generalized Camassa--Holm peakon model. \\ When the deformation parameter tends to $\pm2$, the spectral parameter dependence drops out, and we recover the linear and quadratic Poisson structure of the Camassa--Holm peakon model. \\ When the spectral parameter tends to the fixed points of the involution defining the halving, we recover the Ragnisco--Bruschi deformation of the Camassa--Holm peakon model, thereby establishing its quadratic Poisson structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quadratic Poisson brackets for the Camassa--Holm peakons Avan, J. Frappat, L. Ragoucy, E. Exactly Solvable and Integrable Systems High Energy Physics - Theory Mathematical Physics We establish quadratic Poisson brackets for the generalized Camassa--Holm peakon structure introduced in \cite{AFR23}. The calculation is based on the halving of the spectral parameter dependent $r$-matrix used to define the linear Poisson structure of this model. This quadratic structure, together with the linear one, establish the bi-Hamiltonian structure of the generalized Camassa--Holm peakon model. \\ When the deformation parameter tends to $\pm2$, the spectral parameter dependence drops out, and we recover the linear and quadratic Poisson structure of the Camassa--Holm peakon model. \\ When the spectral parameter tends to the fixed points of the involution defining the halving, we recover the Ragnisco--Bruschi deformation of the Camassa--Holm peakon model, thereby establishing its quadratic Poisson structure. |
| title | Quadratic Poisson brackets for the Camassa--Holm peakons |
| topic | Exactly Solvable and Integrable Systems High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2512.11066 |