Various versions of the Riemann–Hilbert problem for linear differential equations

R. R. Gontsov ; V. A. Poberezhnyi

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 63 (2008) no. 4, pp. 603-639

Résumé

A counterexample to Hilbert's 21st problem was found by Bolibrukh in 1988 (and published in 1989). In the further study of this problem he substantially developed the approach using holomorphic vector bundles and meromorphic connections. Here the best-known results of the past that were obtained by using this approach (both for Hilbert's 21st problem and for certain generalizations) are presented.

Bibliographie

[1] A. A. Bolibrukh, “The Riemann–Hilbert problem”, Russian Math. Surveys, 45:2 (1990), 1–58 | DOI | MR | Zbl

[2] L. Sauvage, “Sur les solutions régulières d'un système d'équations différentielles”, Ann. Sci. École Norm. Sup. (3), 3 (1886), 391–404 | MR | Zbl

[3] Ph. Hartman, Ordinary differential equations, Wiley, New York–London–Sydney, 1964 | MR | MR | Zbl | Zbl

[4] J. Moser, “The order of a singularity in Fuchs' theory”, Math. Z., 72:1 (1959), 379–398 | DOI | MR | Zbl

[5] E. Wagenführer, “On the regularity of systems of linear differential equations at singular points”, Analysis, 4:3–4 (1984), 267–298 | MR | Zbl

[6] D. A. Lutz, R. Schäfke, “On the identification and stability of formal invariants for singular differential equations”, Linear Algebra Appl., 72 (1985), 1–46 | DOI | MR | Zbl

[7] J. Horn, “Zur Theorie der Systeme linearer Differentialgleichungen mit einer unabhängigen Veränderlichen. II”, Math. Ann., 40:4 (1892), 527–550 | DOI | MR | Zbl

[8] A. A. Bolibrukh, “Problema Rimana–Gilberta na kompleksnoi proektivnoi pryamoi”, Matem. zametki, 46:3 (1989), 118–120 | MR | Zbl

[9] D. V. Anosov, A. A. Bolibruch, “The Riemann–Hilbert problem”, Aspects Math., E22, Vieweg, Braunschweig, 1994 | MR | Zbl

[10] A. A. Bolibrukh, Fuksovy differentsialnye uravneniya i izomonodromnye deformatsii (to appear)

[11] V. P. Kostov, “Monodromy groups of regular systems on the Riemann sphere”, Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys., 9, Eur. Math. Soc., Zürich, 2007, 209–254 | MR

[12] J. Plemelj, “Riemannsche Funktionenscharen mit gegebener Monodromiegruppe”, Monatsch. Math. Phys., 19:1 (1908), 211–246 | DOI | MR | Zbl

[13] W. Dekkers, “The matrix of a connection having regular singularities on a vector bundle of rank $2$ on $P^1({\mathbb C})$”, Équations différentielles et systèmes de Pfaff dans le champ complexe (Inst. Rech. Math. Avancée, Strasbourg, 1975), Lecture Notes in Math., 712, Springer, Berlin, 1979, 33–43 | DOI | MR | Zbl

[14] A. A. Bolibrukh, “On sufficient conditions for the positive solvability of the Riemann–Hilbert problem”, Math. Notes, 51:2 (1992), 110–117 | DOI | MR | Zbl

[15] V. P. Kostov, “Fuchsian systems on ${\mathbb C}P^1$ and the Riemann–Hilbert problem”, C. R. Acad. Sci. Paris Sér. I Math., 315:2 (1992), 143–148 | MR | Zbl

[16] A. A. Bolibrukh, “Construction of a Fuchsian equation from a monodromy representation”, Math. Notes, 48:5 (1990), 1090–1099 | DOI | MR | Zbl | Zbl

[17] L. Fuchs, “Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten”, J. Reine Angew. Math., 68 (1868), 354–385 | Zbl

[18] B. Riemann, “Zwei allgemeine Lehrsätze über lineare Differentialgleichungen mit algebraichen Koefficienten”, Math. Werke, 357–369

[19] H. Poincaré, “Sur les groupes des équations linéaires”, Acta Math., 4:1 (1884), 201–312 | DOI | MR | MR | Zbl | Zbl

[20] D. Hilbert, “Mathematische Probleme”, Gött. Nachr., 1900, 253–297 | Zbl

[21] A. A. Bolibruch, “Inverse monodromy problems of the analytic theory of differential equations”, Mathematical events of the twentieth century, Springer, Berlin, 2006, 49–74 | MR | Zbl

[22] D. V. Anosov, V. P. Leksin, “Andrei Andreevich Bolibrukh in life and science (30 January 1950–11 November 2003)”, Russian Math. Surveys, 59:6 (2004), 1009–1028 | DOI | MR | Zbl

[23] I. A. Lappo-Danilevskii, Primenenie funktsii ot matrits k teorii lineinykh sistem obyknovennykh differentsialnykh uravnenii, Gostekhizdat, M., 1957 | MR

[24] A. H. M. Levelt, “Hypergeometric functions. I”, Nederl. Akad. Wetensch. Proc. Ser. A, 64 (1961), 361–372 | MR | Zbl

[25] Yu. S. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations, Grad. Stud. Math., 86, Amer. Math. Soc., Providence, RI, 2008 | MR | Zbl

[26] H. Röhrl, “Das Riemann–Hilbertsche Problem der Theorie der linearen Differentialgleichungen”, Math. Ann., 133:1 (1957), 1–25 | DOI | MR | Zbl

[27] P. Deligne, Equations différentielles à points singuliers réguliers, Lecture Notes in Math., 163, Springer-Verlag, Berlin–New York, 1970 | DOI | MR | Zbl

[28] I. V. V'yugin, R. R. Gontsov, “Additional parameters in inverse monodromy problems”, Sb. Math., 197:12 (2006), 1753–1773 | DOI

[29] O. Forster, Riemannsche Flächen, Heidelberger Taschenbücher, 184, Springer-Verlag, Berlin–New York, 1977 | MR | MR | Zbl

[30] A. A. Bolibruch, “The Riemann–Hilbert problem on a compact Riemannian surface”, Proc. Steklov Inst. Math., 238 (2002), 47–60 | MR | Zbl

[31] I. V. V'yugin, “Constructive solvability conditions for the Riemann–Hilbert problem”, Math. Notes, 77:5–6 (2005), 595–605 | DOI | MR | Zbl

[32] A. A. Bolibruch, “Vector bundles associated with monodromies and asymptotics of Fuchsian systems”, J. Dynam. Control Systems, 1:2 (1995), 229–252 | DOI | MR | Zbl

[33] T. Kimura, K. Shima, “A note on the monodromy of the hypergeometric differential equation”, Japan. J. Math. (N. S.), 17:1 (1991), 137–163 | MR | Zbl

[34] V. A. Poberezhnyi, “Special monodromy groups and the Riemann–Hilbert problem for the Riemann equation”, Math. Notes, 77:5–6 (2005), 695–707 | DOI | MR | Zbl

[35] I. V. Vyugin, “Ob ekvivalentnykh formulirovkakh 21-i problemy Gilberta”, Tezisy dokladov Mezhdunarodnoi konferentsii po differentsialnym uravneniyam i dinamicheskim sistemam (Suzdal), 2008, 62–63

[36] S. Kowalevski, “Sur le problème de la rotation d'un corps solide autour d'un point fixe”, Acta Math., 12:1 (1889), 177–232 | DOI | MR | Zbl

[37] P. Painlevé, “Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme”, Acta Math., 25:1 (1902), 1–85 | DOI | MR | Zbl

[38] R. Fuchs, “Sur quelques équations différentielles linéaires du second ordre”, C. R. Acad. Sci. Paris, 141 (1905), 555–558 | Zbl

[39] B. Gambier, “Sur les équations différentielles du second ordre dont l'intégrale générale est uniforme”, C. R. Acad. Sci. Paris, 142 (1906), 266–269 | Zbl

[40] L. Schlesinger, “Über Losungen gewisser Differentialgleichungen als Funktionen der singularen Punkte”, J. Reine Angew. Math., 129 (1906), 287–294 | Zbl

[41] B. Malgrange, “Sur les déformations isomonodromiques. I. Singularités régulières”, Mathematics and physics, Semin. École Norm. Sup. (Paris, 1979–1982), Progr. Math., 37, Birkhäuser, Boston, MA, 1983, 401–426 | MR | Zbl

[42] M. Jimbo, T. Miwa, “Monodromy preserving deformations of linear differential equations with rational coefficients. II”, Phys. D, 2:3 (1981), 407–448 | DOI | MR

[43] R. Garnier, “Sur des 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 a ses points critiques fixes”, Ann. Sci. École Norm. Sup. (3), 29 (1912), 1–126 | MR | Zbl

[44] P. Painlevé, “Sur les équations différentielles du second ordre à points critiques fixés”, C. R. Acad. Sci. Paris, 143 (1907), 1111–1117 | Zbl

[45] B. Dubrovin, M. Mazzocco, “Canonical structure and symmetries of the Schlesinger equations”, Comm. Math. Phys., 271:2 (2007), 289–373 ; arXiv: math/0311261 | DOI | MR | Zbl

[46] N. J. Hitchin, “Poncelet polygons and the Painlevé equations”, Geometry and analysis (Bombay, 1992), Tata Inst. Fundam. Res., 13, Oxford Univ. Press, Oxford, 1995, 151–185 | MR | Zbl

[47] B. Dubrovin, M. Mazzocco, “Monodromy of certain Painlevé-VI transcendents and reflection groups”, Invent. Math., 141:1 (2000), 55–147 | DOI | MR | Zbl

[48] P. P. Boalch, “From Klein to Painlevé via Fourier, Laplace and Jimbo”, Proc. London Math. Soc. (3), 90:1 (2005), 167–208 | DOI | MR | Zbl

[49] Ph. Boalch, “Six results on Painlevé VI”, Théories asymptotiques et équations de Painlevé, Sémin. Congr., 14, Soc. Math. France, Paris, 2006, 1–20 | MR | Zbl

[50] Ph. Boalch, “Some explicit solutions to the Riemann–Hilbert problem”, Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys., 9, Eur. Math. Soc., Zürich, 2007, 85–112 | MR

[51] H. Esnault, E. Viehweg, “Semistable bundles on curves and irreducible representations of the fundamental group”, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., 241, Amer. Math. Soc., Providence, RI, 1999, 129–138 | MR | Zbl

[52] H. Esnault, C. Hertling, “Semistable bundles on curves and reducible representations of the fundamental group”, Internat. J. Math., 12:7 (2001), 847–855 ; arXiv: math/0101194 | DOI | MR | Zbl

[53] M. Yoshida, “On the number of apparent singularities of the Riemann–Hilbert problem on Riemann surface”, J. Math. Soc. Japan, 49:1 (1997), 145–159 | DOI | MR | Zbl

[54] M. Ohtsuki, “On the number of apparent singularities of a linear differential equation”, Tokyo J. Math., 5:1 (1982), 23–29 | MR | Zbl

[55] A. A. Bolibruch, S. Malek, C. Mitschi, “On the generalized Riemann–Hilbert problem with irregular singularities”, Expo. Math., 24:3 (2006), 235–272 | DOI | MR | Zbl

[56] W. Balser, W. B. Jurkat, D. A. Lutz, “A general theory of invariants for meromorphic differential equations. I. Formal invariants”, Funkcial. Ekvac., 22:2 (1979), 197–221 | MR | Zbl

[57] R. R. Gontsov, I. V. V'yugin, “Some addition to the generalized Riemann–Hilbert problem”, Ann. Fac. Sci. Toulouse (to appear)

[58] W. Balser, “Birkhoff's reduction problem”, Russian Math. Surveys, 59:6 (2004), 1047–1059 | DOI | MR | Zbl

[59] H. L. Turrittin, “Reduction of ordinary differential equations to the Birkhoff canonical form”, Trans. Amer. Math. Soc., 107:3 (1963), 485–507 | DOI | MR | Zbl

[60] V. Poberezhny, “On the Painlevé property of isomonodromic deformations of Fuchsian systems”, Acta Appl. Math., 101:1–3 (2008), 255–263 | DOI | MR

[61] A. Bolibruch, “Inverse problems for linear differential equations with meromorphic coefficients”, Isomonodromic deformations and applications in physics (Montréal, QC, 2000), CRM Proc. Lecture Notes, 31, Amer. Math. Soc., Providence, RI, 2002, 3–25 | MR | Zbl