Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation

Dreyfus, Thomas; Heu, Viktoria

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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 $q{P}_{\mathrm{VI}}$ 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 $1$.

Classification : 14D05, 14F35, 34M56, 39A13

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