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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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
@article{AIF_2024__74_1_121_0, author = {Klime\v{s}, Martin}, title = {Wild monodromy of the {Fifth} {Painlev\'e} equation and its action on wild character variety: an approach of confluence}, journal = {Annales de l'Institut Fourier}, pages = {121--192}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {74}, number = {1}, year = {2024}, doi = {10.5802/aif.3579}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3579/} }
TY - JOUR AU - Klimeš, Martin TI - Wild monodromy of the Fifth Painlevé equation and its action on wild character variety: an approach of confluence JO - Annales de l'Institut Fourier PY - 2024 SP - 121 EP - 192 VL - 74 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3579/ DO - 10.5802/aif.3579 LA - en ID - AIF_2024__74_1_121_0 ER -
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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/
[1] Exponentially small corrections to divergent asymptotic expansions of solutions of the fifth Painlevé equation, Math. Res. Lett., Volume 4 (1997) no. 5, pp. 741-759 | DOI | Zbl
[2] Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer, 2000
[3] Doubly-resonant saddle-nodes in and the fixed singularity at infinity in Painlevé equations: analytic classification, Ann. Inst. Fourier, Volume 68 (2018) no. 4, pp. 1715-1830 | DOI | MR | Zbl
[4] From Klein to Painlevé via Fourier, Laplace, Jimbo, Proc. Lond. Math. Soc., Volume 90 (2005) no. 1, pp. 167-208 | DOI | MR | Zbl
[5] Towards a nonlinear Schwarz’s list, The many facets of geometry: a tribute to Nigel Hitchin (Bourguignon, J.-P.; Garcia-Prada, O.; Salamon, S., eds.), Oxford University Press, 2010, pp. 210-236 | DOI | MR | Zbl
[6] Geometry and braiding of Stokes data; Fission and wild character varieties, Ann. Math., Volume 179 (2014) no. 1, pp. 301-365 | DOI | MR | Zbl
[7] On isomonodromic deformations of Fuchsian systems, J. Dyn. Control Syst., Volume 3 (1997) no. 4, pp. 589-604 | DOI | MR | Zbl
[8] On the classification of cubic surfaces, J. Lond. Math. Soc., Volume 19 (1979), pp. 245-256 | DOI | MR | Zbl
[9] Dynamics on Character Varieties and Malgrange irreducibility of Painlevé VI equation, Ann. Inst. Fourier, Volume 59 (2009) no. 7, pp. 2927-2978 | DOI | Numdam | Zbl
[10] Une preuve galoisienne de l’irréductibilité au sens de Nishioka–Umemura de la première équation de Painlevé, Differential equations and singularities (Astérisque), Volume 324, Société Mathématique de France, 2009, pp. 83-100 | Numdam | MR | Zbl
[11] A memoir on cubic surfaces, Philos. Trans. R. Soc. Lond., Volume 159 (1869), pp. 231-326 | Zbl
[12] Painlevé monodromy manifolds, decorated character varieties, and cluster algebras, Int. Math. Res. Not., Volume 2017 (2017) no. 24, pp. 7639-7691 | Zbl
[13] Geometry of 2D topological field theories, Lecture Notes in Mathematics, 1620, Springer, 1996, pp. 120-348
[14] Painlevé transcendents in two-dimensional topological field theory, The Painlevé property: One century later (Conte, R., ed.) (CRM Series in Mathematical Physics), Springer, 1999, pp. 287-412 | DOI | MR | Zbl
[15] Monodromy of certain Painlevé VI transcendents and reflection groups, Invent. Math., Volume 141 (2000) no. 1, pp. 55-147 | DOI | Zbl
[16] Confluence procedures in the generalized hypergeometric family, J. Math. Sci., Tokyo, Volume 5 (1998) no. 4, pp. 597-625 | MR | Zbl
[17] Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critique fixés, Acta Math., Volume 33 (1910), pp. 1-55 | DOI
[18] Sur les singularités irrégulières des équations différentielles linéaires, Journ. de Math. (8), Volume 2 (1919), pp. 99-200 | Numdam
[19] Stokes Operators via Limit Monodromy of Generic Perturbation, J. Dyn. Control Syst., Volume 5 (1999) no. 1, pp. 101-135 | DOI | MR | Zbl
[20] Confluence of singular points and Stokes phenomena, Normal forms, bifurcations and finiteness problems in differential equations (Ilyashenko, Y.; Rousseau, C., eds.) (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 137, Kluwer Academic Publishers, 2004 | DOI | MR
[21] Middle convolution and deformation for fuchsian systems, J. Lond. Math. Soc., Volume 76 (2007) no. 2, pp. 438-450 | DOI | MR | Zbl
[22] Dual isomonodromic deformations and moment maps to loop algebras, Commun. Math. Phys., Volume 166 (1994) no. 2, pp. 337-365 | DOI | MR | Zbl
[23] Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank , Mosc. Math. J., Volume 14 (2013) no. 2, pp. 309-338 | DOI | Zbl
[24] Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, 86, American Mathematical Society, 2008
[25] Dynamics of the sixth Painlevé equation, Théories asymptotiques et équations de Painlevé – Angers, juin 2004 (Delabaere, É.; Loday-Richaud, M., eds.) (Séminaires et Congrès), Volume 14, Société Mathématique de France, 2006, pp. 103-167 | Zbl
[26] A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad., Ser. A, Volume 78 (2002) no. 7, pp. 131-135 | Zbl
[27] An area-preserving action of the modular group on cubic surfaces and the monodromy of the Painlevé VI equation, Commun. Math. Phys., Volume 242 (2003) no. 1-2, pp. 185-219 | DOI | Zbl
[28] From Gauss to Painlevé: a modern theory of special functions, Aspects of Mathematics, E16, Vieweg & Sohn, 1991 | DOI
[29] Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci., Volume 18 (1982), pp. 1137-1161 | DOI | Zbl
[30] Monodromy preserving deformations of linear ordinary differential equations with rational coefficients II, Physica D, Volume 2 (1981) no. 2, pp. 407-448 | DOI | MR | Zbl
[31] Algebraic analysis of singular perturbation theory, Translations of Mathematical Monographs, 227, American Mathematical Society, 2005 | DOI
[32] An isomonodromy cluster of two regular singularities, J. Phys. A, Math. Gen., Volume 39 (2006) no. 39, pp. 12033-12072 | DOI | MR | Zbl
[33] Stokes phenomenon and confluence in non-autonomous Hamiltonian systems, Qual. Theory Dyn. Syst., Volume 17 (2018), pp. 665-708 | DOI | MR | Zbl
[34] Analytic classification of families of linear differential systems unfolding a resonant irregular singularity, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 16 (2020), 006, 46 pages | DOI | MR | Zbl
[35] Confluence of singularities in hypergeometric systems, Funkc. Ekvacioj, Volume 63 (2020), pp. 153-181 | DOI | MR | Zbl
[36] Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1, Mosc. Math. J., Volume 12 (2012) no. 1, pp. 77-138 | DOI | Zbl
[37] Algebraic solutions of the sixth Painlevé equation, J. Geom. Phys., Volume 85 (2014), pp. 124-163 | DOI | Zbl
[38] Rings of Fricke characters and automorphism groups of free groups, Math. Z., Volume 170 (1980), pp. 91-103 | DOI | MR | Zbl
[39] Le groupoïde de Galois d’un feuilletage, Enseign. Math., Volume 38 (2001) no. 2, pp. 465-501 | Zbl
[40] Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Publ. Math., Inst. Hautes Étud. Sci., Volume 55 (1982), pp. 63-164 | DOI | Numdam | Zbl
[41] Elementary acceleration and multisummability. I, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 54 (1991) no. 4, pp. 331-401 | Numdam | MR | Zbl
[42] Rational solutions of the Painlevé VI equation, J. Phys. A, Math. Gen., Volume 34 (2001), pp. 2281-2294 | DOI
[43] Painlevé sixth equation as isomonodromic deformation equation of an irregular system, Isomonodromic Deformations and Applications in Physics (CRM Proceedings & Lecture Notes), Volume 32, American Mathematical Society, 2002 | MR | Zbl
[44] Affine Weyl Groups, Discrete Dynamical Systems and Painleve Equations, Commun. Math. Phys., Volume 199 (1998) no. 2, pp. 281-295 | DOI | MR | Zbl
[45] Double affine Hecke algebras of rank 1 and affine cubic surfaces, Int. Math. Res. Not., Volume 2004 (2004) no. 18, pp. 877-912 | DOI | MR | Zbl
[46] A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations, J. Phys. A, Math. Gen., Volume 39 (2006) no. 39, pp. 12129-12151 | DOI | Zbl
[47] Sur les feuilletages associés aux équations du second ordre à points critiques fixes de Painlevé, P., Espaces des conditions initiales, Jpn. J. Math., Volume 5 (1979), pp. 1-79 | DOI | Zbl
[48] Polynomial Hamiltonians associated with Painlevé Equations. I, Proc. Japan Acad., Ser. A, Volume 56 (1980), pp. 264-268 | Zbl
[49] Studies on the Painlevé Equations. I. Sixth Painlevé Equation , Ann. Mat. Pura Appl., Volume 146 (1987), pp. 337-381 | DOI | Zbl
[50] Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math., Volume 25 (1902), pp. 1-85 | DOI | MR | Zbl
[51] Confluence de singularités régulières d’équations différentielles en une singularité irrégulière. Modèle de Garnier, Ph. D. Thesis, IRMA Strasbourg (2001) (https://www-irma.u-strasbg.fr/annexes/publications/pdf/01020.pdf)
[52] Dynamics on wild character varieties, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 11 (2015), 068, 21 pages | MR | Zbl
[53] Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier, Volume 59 (2009) no. 7, pp. 2611-2667 | DOI | Numdam | Zbl
[54] Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, 2003 | DOI
[55] Phenomène de Stokes et resommation, C. R. Math. Acad. Sci. Paris, Volume 301 (1985), pp. 99-102 | MR | Zbl
[56] Confluence and resurgence, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 36 (1989), pp. 703-716 | MR | Zbl
[57] Rational Surfaces Associated with Affine Root Systems and Geometry of the Painlevé Equations, Commun. Math. Phys., Volume 220 (2001) no. 1, pp. 165-229 | DOI | MR | Zbl
[58] Über das globale analytische Verhalten der Normallösungen von und zweier Arten von assoziierten Funktionen, Math. Nachr., Volume 121 (1985), pp. 123-145 | DOI | MR | Zbl
[59] Confluence of several regular singular points into an irregular singular one, J. Dyn. Control Syst., Volume 4 (1998) no. 3, pp. 401-424 | DOI | MR | Zbl
[60] Formal fundamental solutions of irregular singular differential equations depending upon parameters, J. Dyn. Control Syst., Volume 7 (2001) no. 4, pp. 501-533 | DOI | MR | Zbl
[61] Analytic integration of some nonlinear ordinary differential equations and the fifth Painlevé equation in the neighborhood of an irregular singular point, Funkc. Ekvacioj, Volume 26 (1983), pp. 301-338 | MR | Zbl
[62] Series Expansions of Painlevé Transcendents near the Point at Infinity, Funkc. Ekvacioj, Volume 58 (2015), pp. 277-319 | DOI | Zbl
[63] A 2-parameter family of solutions of Painlevé equation (V) near the point at infinity, Funkc. Ekvacioj, Volume 26 (1983), pp. 79-113 | MR | Zbl
[64] Reduction for Painlevé equations at the fixed singular points of the second kind, J. Math. Soc. Japan, Volume 42 (1990), pp. 423-443 | MR | Zbl
[65] A general solution of a nonlinear 2-system without Poincaré’s condition at an irregular singular point, Funkc. Ekvacioj, Volume 27 (1984), pp. 367-391 | MR | Zbl
[66] 2-parameter family of solutions of Painlevé equations (I)-(V) at an irregular singular point, Funkc. Ekvacioj, Volume 28 (1985), pp. 233-248 | MR | Zbl
[67] Confluence et phenomène de Stokes, J. Math. Sci., Tokyo, Volume 3 (1996) no. 1, pp. 91-107 | MR | Zbl
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