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Orbifold Hamiltonian Structures of Certain Quasi-Painlevé Equations

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Abstract

For each of two Hamiltonian systems with the quasi-Painlevé property, we establish a holomorphic orbifold Hamiltonian structure for the system on an augmented phase space obtained through blowups. This generalises the orbifold Hamiltonian structure of the first Painlevé equation on its Okamoto space of initial conditions constructed by Iwasaki and Okada. The Hamiltonian structure is provided by an appropriate orbifold atlas for the space and a collection of Hamiltonian functions which are polynomial in coordinates. In the quasi-Painlevé cases we consider, the Hamiltonian structure is with respect to a rational 2-form which is allowed to have certain zeroes, as opposed to holomorphic symplectic forms in the case of the Painlevé equations, which ensures the system is regularisable everywhere on the augmented phase space in a way that corresponds to the quasi-Painlevé property of the equation.

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Funding

GF acknowledges the support of the grant entitled “Geometric approach to ordinary differential equations" (01.03.2023–29.02.2024) funded under New Ideas 3B competition within Priority Research Area III implemented under the “Excellence Initiative - Research University” (IDUB) Programme (University of Warsaw) (nr 01/IDUB/2019/94). The work of GF is also partially supported by the project PID2021-124472NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe". AS is supported by a Japan Society for the Promotion of Science (JSPS) Postdoctoral Fellowship for Research in Japan and also acknowledges the support of JSPS KAKENHI Grant Numbers 21F21775 and 22KF0073.

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland

    Galina Filipuk

  2. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Tokyo, Meguro-ku, 153–8914, Japan

    Alexander Stokes

  3. Waseda Institute for Advanced Study, Waseda University, 1-21-1 Nishiwaseda, Tokyo, Shinjuku-ku, 169–0051, Japan

    Alexander Stokes

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  1. Galina Filipuk
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  2. Alexander Stokes
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GF and AS jointly conceived and carried out the research reported in this manuscript and wrote the text jointly.

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Correspondence to Galina Filipuk.

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Filipuk, G., Stokes, A. Orbifold Hamiltonian Structures of Certain Quasi-Painlevé Equations. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10352-z

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  • DOI: https://doi.org/10.1007/s10884-024-10352-z

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