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no. 5
Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation
Thomas Dreyfus1; Viktoria Heu2
1 Institut de Recherche Mathématique Avancée, U.M.R. 7501 Université de Strasbourg et C.N.R.S. 7, rue René Descartes 67084 Strasbourg, France
2 IRMA, 7 rue René Descartes, 67084 Strasbourg, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 969-1041.

In the current paper we study the q-analogue introduced by Jimbo and Sakai of the well known Painlevé VI differential equation. We explain how it can be deduced from a q-analogue of Schlesinger equations and show that for a convenient change of variables and auxiliary parameters, it admits a q-analogue of Hamiltonian formulation. This allows us to show that Sakai’s q-analogue of Okamoto space of initial conditions for qP VI admits the differential Okamoto space via some natural limit process.

Dans cet article, nous étudions le q-analogue de la sixième équation de Painlevé introduit par Jimbo et Sakai. Nous expliquons comment il peut être retrouvé à partir d’un q-analogue de l’équation de Schlesinger et nous montrons que, après un changement des paramètre, il admet une formulation en terme de q-système Hamiltonien. Cela nous permet nous prouver que le q-analogue de l’espace d’Okamoto des conditions initiales introduit par Sakai admet l’espace d’Okamoto différentiel comme limite lorsque q tend vers 1.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1760
Classification: 14D05, 14F35, 34M56, 39A13

Thomas Dreyfus 1; Viktoria Heu 2

1 Institut de Recherche Mathématique Avancée, U.M.R. 7501 Université de Strasbourg et C.N.R.S. 7, rue René Descartes 67084 Strasbourg, France
2 IRMA, 7 rue René Descartes, 67084 Strasbourg, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Thomas Dreyfus; Viktoria Heu. Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 969-1041. doi : 10.5802/afst.1760. https://afst.centre-mersenne.org/articles/10.5802/afst.1760/

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