Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy
Abstract
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 $g$ nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.13905
 arXiv:
 arXiv:2302.13905
 Bibcode:
 2023arXiv230213905M
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Symplectic Geometry;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 32G34;
 34M55;
 34M56
 EPrint:
 44 pages + appendices. arXiv admin note: text overlap with arXiv:2212.04833