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Marius van der Put; Masa-Hiko Saito
Moduli spaces for linear differential equations and the Painlevé equations
(Espaces de modules pour des équations différentielles linéaires et équations de Painlevé)
Annales de l'institut Fourier, 59 no. 7 (2009), p. 2611-2667, doi: 10.5802/aif.2502
Article: subscription required (your ip address: 54.234.126.92) | Reviews MR 2649335 | Zbl 1189.14021
Class. Math.: 14D20, 14D25, 34M55, 58F05
Keywords: Moduli space for linear connections, irregular singularities, Stokes matrices, monodromy spaces, isomonodromic deformations, Painlevé equations
[1] D. V. Anosov & A. A. Bolibruch, The Riemann-Hilbert problem, Aspects of Mathematics, E22, Friedr. Vieweg & Sohn, 1994 MR 1276272 | Zbl 0801.34002
[2] Philip Boalch, “From Klein to Painlevé via Fourier, Laplace and Jimbo”, Proc. London Math. Soc. (3) 90 (2005) no. 1, p. 167-208
Article | MR 2107041 | Zbl 1070.34123
[3] A. A. Bolibruch, S. Malek & C. Mitschi, “On the generalized Riemann-Hilbert problem with irregular singularities”, Expo. Math. 24 (2006) no. 3, p. 235-272 MR 2250948 | Zbl 1106.34061
[4] H. Flaschka & A. C. Newell, “Monodromy and spectrum preserving deformations. I”, Commun. Math. Phys. 76 (1980), p. 65-116
Article | MR 588248 | Zbl 0439.34005
[5] Robert Fricke & Felix Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, Bände 3 4, Johnson Reprint Corp., 1965 MR 183872 | JFM 32.0430.01
[6] Richard Fuchs, “Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen”, Math. Ann. 63 (1907) no. 3, p. 301-321
Article | MR 1511408 | JFM 38.0362.01
[7] B. Gambier, “Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes”, Acta Math. 33 (1910) no. 1, p. 1-55
Article | MR 1555055 | JFM 40.0377.02
[8] R. Garnier, “Sur les équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale à ses points critiques fixes”, Ann. Ecole Norm. Sup. 29 (1912), p. 1-126
Numdam | MR 1509146 | JFM 43.0382.01
[9] Masuo Hukuhara, “Sur les points singuliers des équations différentielles linéaires. II”, Jour. Fac. Soc. Hokkaido Univ. 5 (1937), p. 123-166, Sur les points singuliers des équations différentielles linéaires. III, Mem. Fac. Sci. Kyushu Univ., 2, (1942), 125–137 Zbl 0016.30502
[10] M. Inaba, “Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence”, Preprint, arXiv:math/0602004, 2006
arXiv
[11] M. Inaba, K. Iwasaki & M.-H. Saito, Dynamics of the sixth Painlevé equation, Théories asymptotiques et équations de Painlevé, Soc. Math. France, 2006, p. 103–167 MR 2353464 | Zbl 1161.34063
[12] M. Inaba, K. Iwasaki & M.-H. Saito, “Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I”, Publ. Res. Inst. Math. Sci. 42 (2006) no. 4, p. 987-1089
Article | MR 2289083 | Zbl 1127.34055
[13] M. Inaba, K. Iwasaki & M.-H. Saito, Moduli of stableparabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II, in Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, Moduli spaces and arithmetic geometry (Tokyo), Adv. Stud. Pure Math., 2006, p. 387-432 Zbl 1115.14005
[14] K. Iwasaki, “An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation”, Comm. Math. Phys. 242 (2003) no. 1-2, p. 185-219 MR 2018272 | Zbl 1044.34051
[15] Katsunori Iwasaki, “A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation”, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002) no. 7, p. 131-135
Article | MR 1930217 | Zbl 1058.34125
[16] Michio Jimbo & Tetsuji Miwa, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Phys. D 2 (1981) no. 3, p. 407-448
Article | MR 625446
[17] Michio Jimbo, Tetsuji Miwa & Kimio Ueno, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau $-function”, Phys. D 2 (1981) no. 2, p. 306-352
Article | MR 630674
[18] B. Malgrange, Sur les déformations isomonodromiques. I. Singularités régulières, Mathematics and physics (Paris, 1979/1982), Birkhäuser Boston, 1983, p. 401–426 MR 728431 | Zbl 0528.32017
[19] Bernard Malgrange, “Déformations isomonodromiques, forme de Liouville, fonction $\tau $”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 5, p. 1371-1392, xiv, xx
Cedram | MR 2127851 | Zbl 1086.34071
[20] Yousuke Ohyama, Hiroyuki Kawamuko, Hidetaka Sakai & Kazuo Okamoto, “Studies on the Painlevé equations. V. Third Painlevé equations of special type $P_{\rm III}(D_7)$ and $P_{\rm III}(D_8)$”, J. Math. Sci. Univ. Tokyo 13 (2006) no. 2, p. 145-204 MR 2277519 | Zbl 1170.34061
[21] Yousuke Ohyama & Shoji Okumura, “A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations”, J. Phys. A 39 (2006) no. 39, p. 12129-12151
Article | MR 2266216 | Zbl 1116.34072
[22] Kazuo Okamoto, “Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé”, Japan. J. Math. (N.S.) 5 (1979) no. 1, p. 1-79 MR 614694 | Zbl 0426.58017
[23] Paul Painlevé, Oeuvres de Paul Painlevé. Tome I, Éditions du Centre National de la Recherche Scientifique, Paris, 1973, Preface by René Garnier, Compiled by Raymond Gérard, Georges Reeb and Antoinette Sec MR 532682 | Zbl 1092.01510
[24] Marius van der Put & Michael F. Singer, Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 328, Springer-Verlag, 2003 MR 1960772 | Zbl 1036.12008
[25] Masa-Hiko Saito, Taro Takebe & Hitomi Terajima, “Deformation of Okamoto-Painlevé pairs and Painlevé equations”, J. Algebraic Geom. 11 (2002) no. 2, p. 311-362
Article | MR 1874117 | Zbl 1022.34079
[26] Masa-Hiko Saito & Hitomi Terajima, “Nodal curves and Riccati solutions of Painlevé equations”, J. Math. Kyoto Univ. 44 (2004) no. 3, p. 529-568
Article | MR 2103782 | Zbl 1117.14015
[27] H. Sakai, “Rational surfaces associated with affine root systems and geometry of the Painlevé equations”, Comm. Math. Phys. 220 (2001), p. 165-229
Article | MR 1882403 | Zbl 1010.34083
[28] Y. Shibuya, “Perturbation of linear ordinary differential equations at irregular singular points”, Funkcial. Ekvac. 11 (1968), p. 235-246
Article | MR 243171 | Zbl 0228.34036
[29] H. Terajima, “Families of Okamoto-Painlevé pairs and Painlevé equations”, Ann. Mat. Pura Appl. (4) 186 (2007) no. 1, p. 99-146
Article | MR 2263893
[30] H. L. Turrittin, “Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point”, Acta Math. 93 (1955), p. 27-66
Article | MR 68689 | Zbl 0064.33603
Marius van der Put; Masa-Hiko Saito
Moduli spaces for linear differential equations and the Painlevé equations
(Espaces de modules pour des équations différentielles linéaires et équations de Painlevé)
Annales de l'institut Fourier, 59 no. 7 (2009), p. 2611-2667, doi: 10.5802/aif.2502
Article: subscription required (your ip address: 54.234.126.92) | Reviews MR 2649335 | Zbl 1189.14021
Class. Math.: 14D20, 14D25, 34M55, 58F05
Keywords: Moduli space for linear connections, irregular singularities, Stokes matrices, monodromy spaces, isomonodromic deformations, Painlevé equations
Résumé - Abstract
A systematic construction of isomonodromic families of connections of rank two on the Riemann sphere is obtained by considering the analytic Riemann–Hilbert map $RH:\mathcal{M} \rightarrow \mathcal{R}$, where $\mathcal{M}$ is a moduli space of connections and $\mathcal{R}$, the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of $RH$ (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces $\mathcal{M}$. The induced Painlevé equations are computed explicitly. Except for the Painlevé VI case, these families have irregular singularities. The analytic classification of irregular singularities yields explicit spaces $\mathcal{R}$, which are families of affine cubic surfaces, related to Okamoto–Painlevé pairs. A weak and a strong form of the Riemann–Hilbert problem is treated. Our paper extends the fundamental work of Jimbo–Miwa–Ueno and is related to recent work on Painlevé equations.
Bibliography
[2] Philip Boalch, “From Klein to Painlevé via Fourier, Laplace and Jimbo”, Proc. London Math. Soc. (3) 90 (2005) no. 1, p. 167-208
Article | MR 2107041 | Zbl 1070.34123
[3] A. A. Bolibruch, S. Malek & C. Mitschi, “On the generalized Riemann-Hilbert problem with irregular singularities”, Expo. Math. 24 (2006) no. 3, p. 235-272 MR 2250948 | Zbl 1106.34061
[4] H. Flaschka & A. C. Newell, “Monodromy and spectrum preserving deformations. I”, Commun. Math. Phys. 76 (1980), p. 65-116
Article | MR 588248 | Zbl 0439.34005
[5] Robert Fricke & Felix Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, Bände 3 4, Johnson Reprint Corp., 1965 MR 183872 | JFM 32.0430.01
[6] Richard Fuchs, “Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen”, Math. Ann. 63 (1907) no. 3, p. 301-321
Article | MR 1511408 | JFM 38.0362.01
[7] B. Gambier, “Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes”, Acta Math. 33 (1910) no. 1, p. 1-55
Article | MR 1555055 | JFM 40.0377.02
[8] R. Garnier, “Sur les équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale à ses points critiques fixes”, Ann. Ecole Norm. Sup. 29 (1912), p. 1-126
Numdam | MR 1509146 | JFM 43.0382.01
[9] Masuo Hukuhara, “Sur les points singuliers des équations différentielles linéaires. II”, Jour. Fac. Soc. Hokkaido Univ. 5 (1937), p. 123-166, Sur les points singuliers des équations différentielles linéaires. III, Mem. Fac. Sci. Kyushu Univ., 2, (1942), 125–137 Zbl 0016.30502
[10] M. Inaba, “Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence”, Preprint, arXiv:math/0602004, 2006
arXiv
[11] M. Inaba, K. Iwasaki & M.-H. Saito, Dynamics of the sixth Painlevé equation, Théories asymptotiques et équations de Painlevé, Soc. Math. France, 2006, p. 103–167 MR 2353464 | Zbl 1161.34063
[12] M. Inaba, K. Iwasaki & M.-H. Saito, “Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I”, Publ. Res. Inst. Math. Sci. 42 (2006) no. 4, p. 987-1089
Article | MR 2289083 | Zbl 1127.34055
[13] M. Inaba, K. Iwasaki & M.-H. Saito, Moduli of stableparabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II, in Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, Moduli spaces and arithmetic geometry (Tokyo), Adv. Stud. Pure Math., 2006, p. 387-432 Zbl 1115.14005
[14] K. Iwasaki, “An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation”, Comm. Math. Phys. 242 (2003) no. 1-2, p. 185-219 MR 2018272 | Zbl 1044.34051
[15] Katsunori Iwasaki, “A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation”, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002) no. 7, p. 131-135
Article | MR 1930217 | Zbl 1058.34125
[16] Michio Jimbo & Tetsuji Miwa, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Phys. D 2 (1981) no. 3, p. 407-448
Article | MR 625446
[17] Michio Jimbo, Tetsuji Miwa & Kimio Ueno, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau $-function”, Phys. D 2 (1981) no. 2, p. 306-352
Article | MR 630674
[18] B. Malgrange, Sur les déformations isomonodromiques. I. Singularités régulières, Mathematics and physics (Paris, 1979/1982), Birkhäuser Boston, 1983, p. 401–426 MR 728431 | Zbl 0528.32017
[19] Bernard Malgrange, “Déformations isomonodromiques, forme de Liouville, fonction $\tau $”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 5, p. 1371-1392, xiv, xx
Cedram | MR 2127851 | Zbl 1086.34071
[20] Yousuke Ohyama, Hiroyuki Kawamuko, Hidetaka Sakai & Kazuo Okamoto, “Studies on the Painlevé equations. V. Third Painlevé equations of special type $P_{\rm III}(D_7)$ and $P_{\rm III}(D_8)$”, J. Math. Sci. Univ. Tokyo 13 (2006) no. 2, p. 145-204 MR 2277519 | Zbl 1170.34061
[21] Yousuke Ohyama & Shoji Okumura, “A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations”, J. Phys. A 39 (2006) no. 39, p. 12129-12151
Article | MR 2266216 | Zbl 1116.34072
[22] Kazuo Okamoto, “Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé”, Japan. J. Math. (N.S.) 5 (1979) no. 1, p. 1-79 MR 614694 | Zbl 0426.58017
[23] Paul Painlevé, Oeuvres de Paul Painlevé. Tome I, Éditions du Centre National de la Recherche Scientifique, Paris, 1973, Preface by René Garnier, Compiled by Raymond Gérard, Georges Reeb and Antoinette Sec MR 532682 | Zbl 1092.01510
[24] Marius van der Put & Michael F. Singer, Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 328, Springer-Verlag, 2003 MR 1960772 | Zbl 1036.12008
[25] Masa-Hiko Saito, Taro Takebe & Hitomi Terajima, “Deformation of Okamoto-Painlevé pairs and Painlevé equations”, J. Algebraic Geom. 11 (2002) no. 2, p. 311-362
Article | MR 1874117 | Zbl 1022.34079
[26] Masa-Hiko Saito & Hitomi Terajima, “Nodal curves and Riccati solutions of Painlevé equations”, J. Math. Kyoto Univ. 44 (2004) no. 3, p. 529-568
Article | MR 2103782 | Zbl 1117.14015
[27] H. Sakai, “Rational surfaces associated with affine root systems and geometry of the Painlevé equations”, Comm. Math. Phys. 220 (2001), p. 165-229
Article | MR 1882403 | Zbl 1010.34083
[28] Y. Shibuya, “Perturbation of linear ordinary differential equations at irregular singular points”, Funkcial. Ekvac. 11 (1968), p. 235-246
Article | MR 243171 | Zbl 0228.34036
[29] H. Terajima, “Families of Okamoto-Painlevé pairs and Painlevé equations”, Ann. Mat. Pura Appl. (4) 186 (2007) no. 1, p. 99-146
Article | MR 2263893
[30] H. L. Turrittin, “Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point”, Acta Math. 93 (1955), p. 27-66
Article | MR 68689 | Zbl 0064.33603
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