Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy
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| Format: | Preprint |
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2023
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| _version_ | 1866909979244494848 |
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| author | Marchal, Olivier Alameddine, Mohamad |
| author_facet | Marchal, Olivier Alameddine, Mohamad |
| contents | In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in $\mathfrak{gl}_2(\mathbb{C})$ admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlevé $1$ hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms of the irregular times and standard $2g$ Darboux coordinates associated to the twisted connection. Furthermore, we obtain a map that reduces the space of irregular times to only $g$ non-trivial isomonodromic deformations. In addition, we perform a symplectic change of Darboux coordinates to obtain a set of symmetric Darboux coordinates in which Hamiltonians and Lax pairs are polynomial. Finally, we apply our general theory to the first cases of the hierarchy: the Airy case $(g=0)$, the Painlevé $1$ case $(g=1)$ and the next two elements of the Painlevé $1$ hierarchy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_13905 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy Marchal, Olivier Alameddine, Mohamad Mathematical Physics High Energy Physics - Theory Symplectic Geometry Exactly Solvable and Integrable Systems 32G34, 34M55, 34M56 In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in $\mathfrak{gl}_2(\mathbb{C})$ admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlevé $1$ hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms of the irregular times and standard $2g$ Darboux coordinates associated to the twisted connection. Furthermore, we obtain a map that reduces the space of irregular times to only $g$ non-trivial isomonodromic deformations. In addition, we perform a symplectic change of Darboux coordinates to obtain a set of symmetric Darboux coordinates in which Hamiltonians and Lax pairs are polynomial. Finally, we apply our general theory to the first cases of the hierarchy: the Airy case $(g=0)$, the Painlevé $1$ case $(g=1)$ and the next two elements of the Painlevé $1$ hierarchy. |
| title | Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy |
| topic | Mathematical Physics High Energy Physics - Theory Symplectic Geometry Exactly Solvable and Integrable Systems 32G34, 34M55, 34M56 |
| url | https://arxiv.org/abs/2302.13905 |