Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations

D. V. Anosov   ; V. P. Leksin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 66 (2011) no. 1, pp. 1-33

Résumé

This paper contains an account of A. A. Bolibrukh's results obtained in the new directions of research that arose in the analytic theory of differential equations as a consequence of his sensational counterexample to the Riemann–Hilbert problem. A survey of results of his students in developing topics first considered by Bolibrukh is also presented. The main focus is on the role of the reducibility/irreducibility of systems of linear differential equations and their monodromy representations. A brief synopsis of results on the multidimensional Riemann–Hilbert problem and on isomonodromic deformations of Fuchsian systems is presented, and the main methods in the modern analytic theory of differential equations are sketched. Bibliography: 69 titles.

Détail

Keywords: regular and Fuchsian systems of linear differential equations, monodromy representations of meromorphic systems of differential equations, Riemann–Hilbert problem, reducible and irreducible monodromy representations and systems of differential equations
Mots-clés : isomonodromic deformations.

Bibliographie

[1] 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

[2] A. A. Bolibrukh, Obratnye zadachi monodromii v analiticheskoi teorii differentsialnykh uravnenii, MTsNMO, M., 2009, 220 pp. http://biblio.mccme.ru/node/2154

[3] R. R. Gontsov, V. A. Poberezhnyi, “Various versions of the Riemann–Hilbert problem for linear differential equations”, Russian Math. Surveys, 63:4 (2008), 603–639 | DOI | MR | Zbl

[4] V. P. Leksin, “A. A. Bolibrukh's works on multidimensional regular and Fuchsian systems”, Russian Math. Surveys, 59:6 (2004), 1155–1164 | DOI | MR | Zbl

[5] A. A. Bolibrukh, “O sistemakh Pfaffa tipa Fuksa”, UMN, 32:2 (1977), 203–204 | MR | Zbl

[6] A. A. Bolibruh, “On the fundamental matrix of a Pfaffian system of Fuchsian type”, Math. USSR-Izv., 11:5 (1977), 1031–1054 | DOI | MR | Zbl

[7] A. A. Bolibrukh, O fundamentalnoi matritse sistem Pfaffa tipa Fuksa, Dis. $\dots$ kand. fiz.-matem. nauk, MGU, M., 1977, 94 pp.

[8] A. A. Bolibrukh, “Primer nerazreshimoi problemy Rimana–Gilberta na $\mathbb{C}P^2$”, Geometricheskie metody v zadachakh algebry i analiza, Izd-vo YarGU, Yaroslavl, 1980, 60–64 | MR

[9] A. H. M. Levelt, “Hypergeometric functions. I”, Indag. Math., 23 (1961), 361–401 | MR | Zbl

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

[11] Yu. Manin, “Moduli Fuchsiani”, Ann. Sc. Norm. Sup. Pisa, 19 (1965), 113–126 | MR | Zbl

[12] P. Deligne, “Équations différentielles à points singuliers réguliers”, Lecture Notes in Math., 163 (1970), Springer-Verlag, Berlin–New York, 133 pp. | DOI | MR | Zbl

[13] N. M. Katz, “An overview of Deligne's work on Hilbert's twenty-first problem”, Mathematical developments arising from Hilbert problems (Northern Illinois Univ., De Kalb, Ill., 1974), Proc. Sympos. Pure Math., 28, Amer. Math. Soc., Providence, RI, 1976, 537–557 | MR | Zbl

[14] D. V. Anosov, Hilbert 21st problem (according to Bolibruch), Preprint No 660, Inst. Math. Appl., Univ. Minnesota, Minneapolis, 1990 http://www.ima.umn.edu/preprints/June90Series/660.pdf

[15] D. V. Anosov, An introduction to Hilbert's 21-st problem, Preprint No 861, Inst. Math. Appl., Univ. Minnesota, Minneapolis, 1991 http://www.ima.umn.edu/preprints/Aug91Series/861.pdf

[16] D. V. Anosov, A. A. Bolibruch, The Riemann–Hilbert problem, Aspects Math., E22, Vieweg, Braunschweig, 1994, 190 pp. | MR | Zbl

[17] F. R. Gantmakher, Teoriya matrits, Gostekhizdat, M., 1953; 4-Рμ РёР·Рґ., Наука, Рњ., 1988, 549 СЃ. ; F. R. Gantmacher, The theory of matrices, 1, 2, Chelsea Publishing Co., New York, 1959 | MR | Zbl | MR

[18] A. A. Bolibrukh, “The Fuchs inequality on a compact Kähler manifold”, Dokl. Math., 64:2 (2001), 213–215 | MR | Zbl

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

[20] J. Plemelj, Problems in the sense of Riemann and Klein, Interscience Tracts in Pure and Applied Mathematics, 16, Wiley, New York–London–Sydney, 1964, vii+175 pp. | MR | Zbl

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

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

[23] A. A. Bolibrukh, The 21st Hilbert problem for linear Fuchsian systems, Proc. Steklov Inst. Math., 206, Amer. Math. Soc., Providence, RI, 1995, viii, 145 pp. | MR | Zbl | Zbl

[24] I. V. V'yugin, “Fuchsian systems with completely reducible monodromy”, Math. Notes, 85:6 (2009), 780–786 | DOI | MR | Zbl

[25] D. V. Anosov, A simple counterexample to the Fuchsian version of the Hilbert 21st problem, Preprint No 124, Max-Planck-Institut für Math., Bonn, 1998 http://www.mpim-bonn.mpg.de/preprints/send?bid=242

[26] D. V. Anosov, “A simple counterexample to the Fuchsian version of the Hilbert 21st problem and related class of monodromy representations”, Ulmer Seminare über Funktionalanalysis und Differentialgleichungen, 3 (1998), 1–16

[27] Yu. S. Il'yashenko, “Three gems in the theory of linear differential equations (in the work of A. A. Bolibrukh)”, Russian Math. Surveys, 59:6 (2004), 1079–1091 | DOI | MR | Zbl

[28] 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

[29] 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

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

[31] A. A. Bolibrukh, “On the existence of Fuchsian systems with given asymptotics”, Proc. Steklov Inst. Math., 216 (1997), 26–37 | MR | Zbl

[32] A. A. Bolibrukh, “On an analytic transformation to standard Birkhoff form”, Dokl. Math., 49:1 (1994), 150–153 | MR | Zbl

[33] A. I. Gladyshev, “On the Riemann–Hilbert problem in dimension 4”, J. Dyn. Control Syst., 6:2 (2000), 219–264 | DOI | MR | Zbl

[34] M. Stephane, “On reducible monodromies realized by reducible Fuchsian systems”, J. Dyn. Control Syst., 5:4 (1999), 509–522 | DOI | MR | Zbl

[35] S. Malek, “Fuchsian systems with reducible monodromy are meromorphically equivalent to reducible Fuchsian systems”, Differentsialnye uravneniya i dinamicheskie sistemy, Sb. statei. K 80-letiyu so dnya rozhdeniya akademika Evgeniya Frolovicha Mischenko, Tr. MIAN, 236, Nauka, M., 2002, 481–490 | MR | Zbl

[36] A. I. Gladyshev, “O privodimykh fuksovykh sistemakh chetvertogo poryadka”, Problemy matematiki v fizicheskikh i tekhnicheskikh zadachakh, MFTI, M., 1994, 66–80

[37] S. Malek, “On Fuchsian systems with decomposable monodromy”, Monodromiya v zadachakh algebraicheskoi geometrii i differentsialnykh uravnenii, Tr. MIAN, 238, Nauka, M., 2002, 196–203 | MR | Zbl

[38] I. V. V'yugin, “An indecomposable Fuchsian system with a decomposable monodromy representation”, Math. Notes, 80:4 (2006), 478–484 | DOI | MR | Zbl

[39] J. Vandamme, “Problème de Riemann–Hilbert pour une représentation de monodromie triangulaire supérieure de dimension 5”, C. R. Acad. Sci. Paris Sér. I Math., 326:9 (1998), 1069–1072 | DOI | MR | Zbl

[40] J. Vandamme, Problème de Riemann–Hilbert pour une représentation de monodromie triangulaire supérieure, Thèse de doctorat en Sciences Mathématiques, Université de Nice-Sophia Antipolis, UFR Faculté des Sciences, Lab. Mathématiques J. A. Dieudonné, 25 mai 1998, 101 pp.

[41] M. A. Butuzov, “Ob usloviyakh razreshimosti problemy Rimana–Gilberta v klasse verkhnetreugolnykh sistem”, Sbornik nauchnykh statei aspirantov i soiskatelei, 80, no. 4, KGPI, Kolomna, 2006, 193–198

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

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

[44] I. V. Vyugin, “Postroenie fuksovoi sistemy po fuksovu uravneniyu”, Vestnik KGPI: Matematicheskie i estestvennye nauki, 2:3 (2007), 43–47

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

[46] V. I. V'yugin, “Hilbert's 21st problem for scalar Fuchsian equations”, Dokl. Math., 79:2 (2009), 203–206 | DOI | MR | Zbl

[47] 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

[48] R. R. Gontsov, I. V. Vyugin, “Some addition to the generalized Riemann–Hilbert problem”, Ann. Fac. Sci. Toulouse Math. (6), 18:3 (2009), 561–576 | MR | Zbl

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

[50] A. A. Bolibrukh, “Meromorphic transformation to Birkhoff standard form in small dimensions”, Proc. Steklov Inst. Math., 225 (1999), 78–86 | MR | Zbl

[51] A. D. Bruno, “Meromorphic reducibility of a linear triangular system of ordinary differential equations”, Dokl. Math., 61:2 (2000), 243–246 | MR | Zbl

[52] A. A. Bolibrukh, “The minimal Birkhoff standard form and the reducibility type”, Differ. Equ., 39:6 (2003), 759–766 | DOI | MR | Zbl

[53] A. A. Bolibruch, “On isomonodromic deformations of Fuchsian systems”, J. Dyn. Control Syst., 3:4 (1997), 589–604 | DOI | MR | Zbl

[54] D. V. Anosov, “Concerning the definition of isomonodromic deformation of Fuchsian systems”, Ulmer Seminare über Funktionalanalysis und Differentialgleichungen, 2 (1997), 1–12

[55] V. A. Poberezhnyi, “General linear problem of the isomonodromic deformation of Fuchsian systems”, Math. Notes, 81:3–4 (2007), 529–542 | DOI | MR | Zbl

[56] A. A. Bolibrukh, “On the tau function for the Schlesinger equation of isomonodromic deformations”, Math. Notes, 74:2 (2003), 177–184 | DOI | MR | Zbl

[57] R. R. Gontsov, “On solutions of the Schlesinger equation in the neigborhood of the Malgrange $\Theta$-divisor”, Math. Notes, 83:5 (2008), 707–711 | DOI | MR | Zbl

[58] A. A. Bolibrukh, “Regular singular points as isomonodromic confluences of Fuchsian singular points”, Russian Math. Surveys, 56:4 (2001), 745–746 | DOI | MR | Zbl

[59] Yu. P. Bibilo, “Isomonodromic confluence of singular points”, Math. Notes, 87:3–4 (2010), 309–315 | DOI | Zbl

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

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

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

[63] A. A. Ryabov, “Splitting types of vector bundles constructed from the monodromy of a given Fuchsian system”, Comput. Math. Math. Phys., 44:4 (2004), 640–648 | MR | Zbl

[64] E. Corel, “Inégalités de Fuchs pour les systèmes différentiels réguliers”, C. R. Acad. Sci. Paris Sér. I Math., 328:11 (1999), 983–986 | DOI | MR | Zbl

[65] E. Corel, “Relations de Fuchs pour les systèmes différentiels irréguliers”, C. R. Acad. Sci. Paris Sér. I Math., 333:4 (2001), 297–300 | DOI | MR | Zbl

[66] E. Corel, “Exponents of a meromorphic connection on a compact Riemann surface”, Pacific J. Math., 242:2 (2009), 259–279 | DOI | MR | Zbl

[67] R. R. Gontsov, “Orders of zeros of polynomials on trajectories of solutions of a system of linear differential equations with regular singular points”, Math. Notes, 76:3 (2004), 438–442 | DOI | MR | Zbl

[68] R. R. Gontsov, “Refined Fuchs inequalities for systems of linear differential equations”, Izv. Math., 68:2 (2004), 259–272 | DOI | MR | Zbl

[69] A. A. Bolibrukh, “Multiplicities of zeros of the components of solutions of a system with regular singular points”, Proc. Steklov Inst. Math., 236 (2002), 53–57 | MR | Zbl