Wild monodromy of the Fifth Painlevé equation and its action on wild character variety: an approach of confluence
Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 121-192.

The article studies the nonlinear Stokes phenomenon at the irregular singularity of the Fifth Painlevé equation from the point of view of confluence from the Sixth Painlevé equation. This approach is developed separately on both sides of the Riemann–Hilbert correspondence. On the side of the Painlevé–Okamoto foliation, the relation between the nonlinear monodromy group of Painlevé VI and the “nonlinear wild monodromy pseudogroup” of Painlevé V (the pseudogroup generated by nonlinear Stokes operators and nonlinear exponential torus) is explained. On the side of the associated linear isomonodromic problems, the “wild” character variety (the space of the linear monodromy and Stokes data) of Painlevé V is constructed through a birational transformation from the one of Painlevé VI. Explicit formulas for the action of the “nonlinear wild monodromy” of Painlevé V on its character variety are then obtained by transporting the description of the action of the nonlinear monodromy of Painlevé VI on its character variety to that of Painlevé V.

L’article étudie le phénomène de Stokes non linéaire à la singularité irrégulière de la Cinquième équation de Painlevé du point de vue de la confluence à partir de la Sixième équation de Painlevé. Cette approche est développée séparément des deux côtés de la correspondance de Riemann–Hilbert. Du côté du feuilletage de Painlevé–Okamoto, la relation entre le groupe de monodromie non-linéaire de Painlevé VI et le « pseudogroupe de monodromie sauvage non-linéaire » de Painlevé V (le pseudogroupe engendré par les opérateurs de Stokes non-linéaires et le tore exponentiel non-linéaire) est expliquée. Du côté des problèmes isomonodromiques linéaires associés, la variété de caractères « sauvages » (l’espace de la monodromie linéaire et des données de Stokes) de Painlevé V est construite par une transformation birationnelle à partir de celle de Painlevé VI. On obtient alors des formules explicites de l’action de la « monodromie sauvage non-linéaire » de Painlevé V sur sa variété de caractères en transportant la description de l’action de la monodromie non-linéaire de Painlevé VI sur sa variété de caractères à celle de Painlevé V.

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DOI: 10.5802/aif.3579
Classification: 34M40, 34M55, 34M56
Keywords: Painlevé equations, wild character variety, confluence, nonlinear Stokes phenomenon
Mot clés : équations de Painlevé, variété de caractères sauvages, confluence, phénomène de Stokes non-linéaire
Klimeš, Martin 1

1 Department of Applied Mathematics Faculty of Electrical Engineering and Computing University of Zagreb (Croatia)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Klimeš, Martin. Wild monodromy of the Fifth Painlevé equation and its action on wild character variety: an approach of confluence. Annales de l'Institut Fourier, Volume 74 (2024) no. 1, pp. 121-192. doi : 10.5802/aif.3579. https://aif.centre-mersenne.org/articles/10.5802/aif.3579/

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