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arXiv · 2209.10515

Takasaki's rational fourth Painlevé-Calogero system and geometric regularisability of algebro-Painlevé equations

Galina Filipuk·Alexander Stokes
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Abstract

We study a Hamiltonian system without the Painlevé property and show that it admits a kind of regularisation on a bundle of rational surfaces with certain divisors removed, generalising Okamoto's spaces of initial conditions for the Painlevé differential equations. The system in question was obtained by Takasaki as part of the Painlevé-Calogero correspondence and possesses the algebro-Painlevé property, being related by an algebraic transformation to the fourth Painlevé equation. We provide an atlas for the bundle of surfaces in which the system has a global Hamiltonian structure, with all Hamiltonian functions being polynomial in coordinates just as in the case of Okamoto's spaces. We compare the surface associated with the Takasaki system with that of the fourth Painlevé equation, showing that they are related by a combination of blowdowns and a branched double cover, under which we lift the birational Bäcklund transformation symmetries of the fourth Painlevé equation to algebraic ones of the Takasaki system, including a discrete Painlevé equation. We also discuss and provide more examples in support of the idea that there is a connection between the algebro-Painlevé property and similar notions of regularisability, in an analogous way to how regular initial value problems for the Painlevé equations everywhere on Okamoto's spaces are related to the Painlevé property.

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Galina Filipuk, Alexander Stokes. 2022-09-21. Takasaki's rational fourth Painlevé-Calogero system and geometric regularisability of algebro-Painlevé equations. https://arxiv.org/abs/2209.10515

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