Skip to main content
SpringerLink
Log in

Inverse Monodromy Problems of the Analytic Theory of Differential Equations

  • Chapter
Mathematical Events of the Twentieth Century

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. D.V. Anosov, A.A. Bolibruch. The Riemann-Hilbert Problem. Braunschweig: Vieweg & Sohn, 1994 (Aspects Math., E22).

    MATH  Google Scholar 

  2. V. I. Arnold, Yu. S. Il’yashenko. Ordinary differential equations. Ordinary differential equations. In: Ordinary Differential Equations and Smooth Dynamical Systems (eds. D. V. Anosov and V. I. Arnold). Berlin: Springer, 1988, 1–148 (Dynamical Systems I; Encyclopædia Math. Sci., 1).

    Google Scholar 

  3. W. Balser. Meromorphic transformation to Birkhoff standard form in dimension three. J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 1989, 36, 233–246.

    MATH  MathSciNet  Google Scholar 

  4. W. Balser. Analytic transformation to Birkhoff standard form in dimension three. Funkcialaj Ekvacioj, 1990, 33(1), 59–67.

    MATH  MathSciNet  Google Scholar 

  5. W. Balser, A.A. Bolibruch. Transformation of reducible equations to Birkhoff standard form. In: Ulmer Seminare über Funktionalanalysis und Differentialgleichungen, Heft 2, 1997, 73–81.

    Google Scholar 

  6. W. Balser, W. B. Jurkat, D.A. Lutz. Invariants for reducible systems of meromorphic differential equations. Proc. Edinburgh Math. Soc., 1980, 23, 163–186.

    Article  MATH  MathSciNet  Google Scholar 

  7. G.D. Birkhoff. A theorem on matrices of analytic functions. Math. Ann., 1913, 74, 122–133.

    Article  MathSciNet  Google Scholar 

  8. G.D. Birkhoff. The generalized Riemann problem for linear differential equations. Proc. Amer. Acad. Arts Sci., 1913, 33(1), 531–568.

    Google Scholar 

  9. G.D. Birkhoff. Collected Mathematical Papers, Vol. 1. New York: Dover, 1968.

    MATH  Google Scholar 

  10. A.A. Bolibruch. On the fundamental matrix of a Pfaffian system of Fuchsian type. Math. USSR, Izv., 1977, 11, 1031–1054.

    Article  Google Scholar 

  11. A.A. Bolibruch. Pfaffian systems of Fuchs type on a complex analytic manifold. Math. USSR, Sb., 1977, 32, 98–108.

    Article  Google Scholar 

  12. A.A. Bolibruch. An example of an unsolvable Riemann-Hilbert problem on ℂP 2. In: Geometric Methods in Problems of Algebra and Analysis, No. 2. Yaroslavl’ State University, 1980, 60–64 (Russian).

    Google Scholar 

  13. A.A. Bolibruch. The Riemann-Hilbert problem. Russ. Math. Surveys, 1990, 45(2), 1–58.

    Article  Google Scholar 

  14. A.A. Bolibruch. On sufficient conditions for the positive solvability of the Riemann-Hilbert problem. Math. Notes, 1992, 51(2), 110–117.

    Article  MathSciNet  Google Scholar 

  15. A.A. Bolibruch. On analytic transformation to standard Birkhoff form. Russ. Acad. Sci. Dokl. Math., 1994, 49(1), 150–153.

    Google Scholar 

  16. A.A. Bolibruch. The 21st Hilbert problem for linear Fuchsian systems. Proc. Steklov Inst. Math., 1994, 206, viii + 145 pp.

    Google Scholar 

  17. A.A. Bolibruch. On the Birkhoff standard form of a linear system of ODE. In: Mathematics in St. Petersburg. Providence, RI: Amer. Math. Soc., 1996, 169–179 (AMS Transl., Ser. 2, 174).

    Google Scholar 

  18. A.A. Bolibruch. Meromorphic transformation to Birkhoff standard form in small dimensions. Proc. Steklov Inst. Math., 1999, 225, 78–86.

    Google Scholar 

  19. A.A. Bolibruch. On sufficient conditions for the existence of a Fuchsian equation with prescribed monodromy. J. Dynam. Control Systems, 1999, 5(1), 453–472.

    Article  MathSciNet  Google Scholar 

  20. A.A. Bolibruch. On Fuchsian systems with given asymptotics and monodromy. Proc. Steklov Inst. Math., 1999, 224, 98–106.

    MATH  Google Scholar 

  21. A.A. Bolibruch. On orders of movable poles of the Schlesinger equation. J. Dynam. Control Systems, 2000, 6(1), 57–74.

    Article  MATH  MathSciNet  Google Scholar 

  22. A.A. Bolibruch. Fuchsian Differential Equations and Holomorphic Fiber Bundles. Moscow Center for Continuous Mathematical Education, 2000 (Russian).

    Google Scholar 

  23. A.A. Bolibruch. Inverse problems for linear differential equations with meromorphic coefficients. In: Isomonodromic Deformations and Applications in Physics (Montréal, 2000). Providence, RI: Amer. Math. Soc., 2002, 3–25 (CRM Proc. & Lecture Notes, 31).

    Google Scholar 

  24. A.A. Bolibruch. Stable vector bundles with logarithmic connections and the Riemann-Hilbert problem. Russ. Acad. Sci. Dokl. Math., 2001, 64(3), 298–301.

    MATH  Google Scholar 

  25. A.D. Bryuno. The meromorphic reducibility of a linear triangular ODE system. Russ. Acad. Sci. Dokl. Math., 2000, 61(2), 243–247.

    Google Scholar 

  26. D.G. Babbitt, V. S. Varadarajan. Local moduli for meromorphic differential equations. Astérisque, 1989, 169–170, 217 pp.

    Google Scholar 

  27. W. Dekkers. The matrix of a connection having regular singularities on a vector bundle of rank 2 on P 1(ℂ). In: Équations différentielles et systèmes de Pfaff dans le champ complexe (Strasbourg, 1975). Berlin: Springer, 1979, 33–43 (Lecture Notes in Math., 712).

    Google Scholar 

  28. P. Deligne. Équations différentielles à points singuliers réguliers. Berlin — New York: Springer, 1970 (Lecture Notes in Math., 163).

    MATH  Google Scholar 

  29. V.A. Dobrovol’skii. Essays on the Development of the Analytic Theory of Differential Equations. Kiev: Vyshcha Shkola, 1974 (Russian).

    Google Scholar 

  30. H. Esnault, C. Herling. Semistable bundles on curves and reducible representations of the fundamental group. Internat. J. Math., 2001, 12(7), 847–855; http://www.arXiv.org/abs/math.AG/0101194

    Article  MATH  MathSciNet  Google Scholar 

  31. N. P. Erugin. The Riemann Problem. Minsk: Nauka i Tekhnika, 1982 (Russian).

    MATH  Google Scholar 

  32. H. Esnault, E. Viehweg. Semistable bundles on curves and irreducible representations of the fundamental group. In: Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998). Providence, RI: Amer. Math. Soc., 1999, 129–138 (Contemp. Math., 241).

    Google Scholar 

  33. O. Forster. Lectures on Riemann Surfaces, 2nd edition. Berlin: Springer, 1981.

    MATH  Google Scholar 

  34. H. Flaschka, A. C. Newell. Monodromy and spectrum preserving deformations. Commun. Math. Phys., 1980, 76, 67.

    Article  MathSciNet  Google Scholar 

  35. F. R. Gantmacher. Theory of Matrices (transl. K.A.Hirsch). New York: Chelsea, 1959.

    MATH  Google Scholar 

  36. R. Gérard. Le problème de Riemann-Hilbert sur une variété analytique complexe. Ann. Inst. Fourier (Grenoble), 1977, 24(9), 357–412.

    Google Scholar 

  37. V.A. Golubeva. Some problems in the analytic theory of Feynman integrals. Russ. Math. Surveys, 1976, 31(2), 139–207.

    Article  MATH  MathSciNet  Google Scholar 

  38. V.A. Golubeva. On the recovery of a Pfaffian system of Fuchsian type from the generators of the monodromy group. Math. USSR, Izv., 1981, 17, 227–241.

    Article  MATH  Google Scholar 

  39. P. Hartman. Ordinary Differential Equations, 2nd edition. Boston, MA: Birkhäuser, 1982.

    MATH  Google Scholar 

  40. R.M. Hain. On a generalization of Hilbert’s 21st problem. Ann. Sci. École Norm. Supér. (4), 1986, 19(4), 609–627.

    MATH  MathSciNet  Google Scholar 

  41. D. Hilbert. Gesammelte Abhandlungen. Bronx, NY: Chelsea, 1965.

    Google Scholar 

  42. Yu. S. Il’yashenko. Riemann-Hilbert problem and box dimension of attractors, to appear.

    Google Scholar 

  43. Yu. S. Il’yashenko. The nonlinear Riemann-Hilbert problem. Proc. Steklov Inst. Math., 1996, 213, 6–29.

    MathSciNet  Google Scholar 

  44. A. Its, V. Novokshenov. The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Berlin: Springer, 1986 (Lecture Notes in Math., 1191).

    MATH  Google Scholar 

  45. K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida. From Gauss to Painlevé. A Modern Theory of Special Functions. Braunschweig: Vieweg & Sohn, 1991 (Aspects Math., E16).

    MATH  Google Scholar 

  46. W. B. Jurkat, D.A. Lutz, A. Peyerimhoff. Birkhoff invariants and effective calculations for meromorphic differential equations, I; II. J. Math. Anal. Appl., 1976, 53, 438–470; Houston J. Math., 1976, 2, 207–238.

    Article  MATH  MathSciNet  Google Scholar 

  47. M. Jimbo, T. Miwa. Monodromy preserving deformations of linear ordinary differential equations, I; II. Physica D, 1981, 2, 306–352; 407–448.

    Article  MathSciNet  Google Scholar 

  48. N.M. Katz. An overview of Deligne’s work on Hilbert’s twenty-first problem. Proc. Sympos. Pure Math., 1976, 28, 537–559.

    MATH  Google Scholar 

  49. M. Kita. The Riemann-Hilbert problem and its applications to analytic functions of several complex variables. Tokyo J. Math., 1979, 2(1), 1–27.

    Article  MATH  MathSciNet  Google Scholar 

  50. A. Treibich Kohn. Un résultat de Plemelj. In: Mathematics and Physics (Paris, 1979–1982). Boston: Birkhäuser, 1983, 307–312 (Progr. Math., 37).

    Google Scholar 

  51. V. P. Kostov. Fuchsian systems on CP 1 and the Riemann-Hilbert problem. C. R. Acad. Sci. Paris, Sér. I Math., 1992, 315, 143–148.

    MATH  MathSciNet  Google Scholar 

  52. V. P. Kostov. Stratification of the space of monodromy groups of Fuchsian linear systems on ℂP 1. In: Complex Analytic Methods in Dynamical Systems (IMPA, January 1992). Astérisque, 1994, 222, 259–283.

    MATH  MathSciNet  Google Scholar 

  53. B. L. Krylov. Solution in finite form of the Riemann problem for a Gaussian system. Trudy Kazan. Aviats. Inst., 1956, 31, 203–445 (Russian).

    Google Scholar 

  54. I.A. Lappo-Danilevskii. Application of Functions of Matrices to the Theory of Linear Systems of Ordinary Differential Equations. Moscow: GTTI (State Technical-Theoretic Publishing House), 1957 (Russian).

    Google Scholar 

  55. A.H.M. Levelt. Hypergeometric functions. Nederl. Akad. Wetensch. Proc., Ser. A, 1961, 64, 361–401.

    MATH  MathSciNet  Google Scholar 

  56. V. P. Leksin. Meromorphic Pfaffian systems on complex projective spaces. Math. USSR., Sb., 1987, 57, 211–227.

    Article  Google Scholar 

  57. V. P. Leksin. On Fuchsian representations of the fundamental group of complex manifolds. In: Qualitative and Approximate Methods of Studying Operator Equations. Yaroslavl’ State University, 1979, 109–114 (Russian).

    Google Scholar 

  58. B. Malgrange. Sur les déformations isomonodromiques. I. Singularités régulières. In: Mathematics and Physics (Paris, 1979–1982). Boston, MA: Birk-häuser, 1983, 401–426 (Progr. Math., 37).

    Google Scholar 

  59. B. Malgrange. Sur les déformations isomonodromiques. II. Singularités irrégulières. In: Mathematics and Physics (Paris, 1979–1982). Boston, MA: Birk-häuser, 1983, 427–438 (Progr. Math., 37).

    Google Scholar 

  60. S. Malek. Systèmes fuchsiens à monodromie réductible. C. R. Acad. Sci. Paris, Sér. I Math., 2001, 332, 691–694.

    MATH  MathSciNet  Google Scholar 

  61. C. Okonek, M. Schneider, H. Spindler. Vector Bundles on Complex Projective Spaces. Boston, MA: Birkhäuser, 1980.

    MATH  Google Scholar 

  62. J. Palmer. Zeros of the Jimbo, Miwa, Ueno tau function. J. Math. Phys., 1999, 40(12), 6638–6681.

    Article  MATH  MathSciNet  Google Scholar 

  63. J. Plemelj. Problems in the Sense of Riemann and Klein. New York: Wiley Interscience, 1964.

    MATH  Google Scholar 

  64. H. Poincaré. Sur les groupes des équations linéaires. Acta Math., 1883, 4, 201–312.

    Article  Google Scholar 

  65. B. Riemann. Gesammelte mathematische Werke (ed. R. Narasimhan). Berlin — New York: Springer, 1990.

    Google Scholar 

  66. H. Röhrl. Das Riemann-Hilbertsche Problem der Theorie der linearen Differentialgleichungen. Math. Ann., 1957, 133, 1–25.

    Article  MATH  MathSciNet  Google Scholar 

  67. C. Sabbah. Frobenius manifolds: isomonodromic deformations and infinitesimal period mappings. Expos. Math, 1998, 16(1), 1–57.

    MATH  MathSciNet  Google Scholar 

  68. L. Schlesinger. Über Lösungen gewisser Differentialgleichungen als Funktionen der singulären Punkte. J. Reine Angew. Math., 1905, 129, 287–294.

    Google Scholar 

  69. Y. Sibuya. Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation. Providence, RI: Amer. Math. Soc., 1990 (Transl. Math. Monographs, 82).

    MATH  Google Scholar 

  70. O. Suzuki. The problems of Riemann and Hilbert and the relations of Fuchs in several complex variables. In: Équations différentielles et systèmes de Pfaff dans le champ complexe (Strasbourg, 1975). Berlin: Springer, 1979, 325–364 (Lecture Notes in Math., 712).

    Google Scholar 

  71. H. L. Turrittin. Reduction of ordinary differential equations to the Birkhoff canonical form. Trans. Amer. Math. Soc., 1963, 107, 485–507.

    Article  MATH  MathSciNet  Google Scholar 

  72. W. Wasow. Asymptotic Expansions for Ordinary Differential Equations. Huntington, NY: Robert E. Krieger, 1976; reprinted by Dover, 1987.

    MATH  Google Scholar 

  73. M. Yoshida, K. Takano. Local theory of Fuchsian systems. Proc. Japan Acad., 1975, 51(4), 219–223.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. A. A. Bolibruch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Academy of Sciences, Leninsky Pr. 14, 117901, Moscow, Russia

    Yu. S. Osipov

  2. Department of Mathematics, University of Princeton, Washington Road, 08544-1000, Princeton, USA

    Ya. G. Sinai

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bolibruch, A.A. (2006). Inverse Monodromy Problems of the Analytic Theory of Differential Equations. In: Bolibruch, †.A.A., et al. Mathematical Events of the Twentieth Century. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-29462-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation