2022/09/21 by Galina Filipuk, Filipuk, Galina, Alexander Stokes +1
Mathematics · Physics and Astronomy · #34M55 #Advanced Topics in Algebra #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2209.10515
openalex publication_date 2022/09/21 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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.