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Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I

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Current Trends in Operator Theory and its Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 149))

Abstract

In the present paper we discuss the general facts concerning the Schlesinger system: the 7-function, the local factorization of solutions of Fuchsian equations and holomorphic deformations. We introduce the terminology “isoprincipal” for the deformations of Fuchsian equations with general (not necessarily non-resonant) matrix coefficients corresponding to solutions of the Schlesinger system. Every isoprincipal deformation is isomonodromic. The converse is also true in the non-resonant case, but not in general.

In the forthcoming sequel we shall give an explicit description of a class of rational solutions of the Schlesinger system, based on the techniques developed here, and the realization theory for rational matrix functions.

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The first author was supported by the Minerva foundation.

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References

  1. Ablowitz, M., Kaup, D., Newell, A. and H. Segur. Nonlinear-evolution equations of physical significance. Phys. Rev. Let., 32:2, (1973), 125–127.

    MathSciNet  Google Scholar 

  2. Anosov, D.V. and A.A. Bolibruch. The Riemann-Hilbert Problem. (Aspects of Mathematics, Vol. E 22), Vieveg, Braunschweig • Wiesbaden, 1994, i-ix + 190 pp.

    Google Scholar 

  3. Aphojib,A, Ko.abryo nozo.rao.aozuú 2pynnba npainaerbl.x xoc. MaTeM. 3a- McTxH, 5:2, (1969), c. 227–231. Engl. transi.: ARNOL’D, V.I. The cohomology ring of the colored braid group. Math. Notes, 5, pp. 138–140 (1969).

    Google Scholar 

  4. ] Birkhoff, G. D. The generalized Riemann problem for linear differential equa-tions and the allied problems for linear difference and q-difference equations. Proc. Amer. Acad. Arts and Sci., 49 (1913)

    Google Scholar 

  5. Pojihbpyx, A. A. 17po6üae.o’ta Pu.ntana-Tu.ab6epma. Ycnexn MaTeM. xayx, 45:2(1990), c. 3–47. Engl. transi.: BOLIBRUCH, A.A. The Riemann-Hilbert problem. Russian Math. Surveys, 45:2 (1990), pp. 1–47.

    Google Scholar 

  6. Bojihbpyx, A. A. 21-sr npo6.ae.ma Fu.ab6epma d.asr Oyxcoebax üucneiiubix cucme.n-a. TpyALI Matem. Hhcthtyta HM. B.A. CTeKJIOna, 206. Moczsa, Hayxa, 1994., 160 c. Engl. transi.: Bolibruch, A.A. Hilbert’s twenty first problem for Fuchsian systems. Proc. Steklov Inst. Math, 5 (206), 1995, viii + 145 pp.

    Google Scholar 

  7. Bojihbpyx, A. A. ‘yxcoeba,LjuNepentyua.abubae Ypaeneuusr u Po.aomopßnue Pacc.aoenusr. (In Russian): Bolibruch, A.A. Fuchsian Differencial Equations and Holomorphic Bundles. MLIHMO (hI3,AaTe.rzbcTSo Mocxoncxoro IIeazpa Henpepbmnoro MazeMaTnueczoro o6pa3osaxna.), Moscow, 2000, 120 pp.

    Google Scholar 

  8. Boutet DE Monvel, L., A.Douady and J.-L. Verdier - editors. Mathématique et Physique. Séminaire de l’Ecole Normale Supériore 1979–1982. (Progress in Mathematics, vol. 37). Birkhäuser, Boston • Basel • Stuttgart, 1983.

    Google Scholar 

  9. Coddington, E.A. and N. LEVINSON. Theory of Ordinary Differential Equations. McGraw Hill, New York•Toronto•London, 1955.

    MATH  Google Scholar 

  10. Deift, P., ITS, A., Kapaev, A. and X. Zhou: On the Algebro-Geometric Integration of the Schlesinger Equations. Commun. Math. Phys., 203 (1999), pp. 613–633.

    Article  MathSciNet  MATH  Google Scholar 

  11. Dickey, L.A. Soliton Equations and Hamiltonian Systems. World Scientific. Singapure•New Jersey•London•Hong Kong, First Edition (Advanced Series in Mathematical Physics, Vol. 12) - 1991, ix+310 pp.; Second Edition (Advanced Series in Mathematical Physics, Vol. 26) - xii+420 pp., 2003.

    Google Scholar 

  12. ] Date, E., M. Jimbo, M. KASHIWARA, and T. Miwa. Transformation groups for soliton equations. Proc. Japan. Acad. Ser. A Math. Sci.: I. 53:1 (1977), pp. 6–10; II. 53:5 (1977), pp. 147–152; III. 53:5 (1977), pp. 153–158; IV. 53:6 (1977), pp. 183–185; V. 53:7 (1977), pp. 219–224; VI. 54:1 (1978), pp. 1–5; VII. 54:2 (1978), pp. 36–41.

    Google Scholar 

  13. Date, E., M. Jimbo, M. Kashiwara, and T. Miwa. Solitons, T functions and Euclidean Lie algebras. Pp. 261–278 in: Mathématique and Physique. Séminaire de l’Ecole Normale Supérieure 1979–1982. Boutet De Monvel, L., A. Douady, and J.-L. Verdier - eds. (Progress in Math., 37). Birkhäuser, Boston•Basel•Stuttgart, 1983

    Google Scholar 

  14. ] Fuchs, L. Gesammelte Mathematische Werke. Band 3. (Herausgegeben by R. FUCHS und L Schlesinger). Mayer & Müller, Berlin, 1909.

    Google Scholar 

  15. [15] Fuchs, L. Zur Theorie der linearen Differentialgleichungen. Sitzungsberichte der K. preuss. Akademie der Wissenschaften zu Berlin. Einleitung und No. 1–7, 1888, S. 1115–1126; No. 8–15, 1888, S.1273–1290; No. 16–21, 1889, S. 713–726; No. 22–31, 1890, S. 21–38. Reprinted in: [FuL], S. 1–68.

    Google Scholar 

  16. [16] Fuchs, L. Über lineare Differentialgleichungen, welche von Parametern unabhängige substitutionsgruppen besitzen. Sitzungsberichte der K. preuss. Akademie der Wissenschaften zu Berlin. 1892, S. 157–176. Reprinted in: [FuL], S. 117–139.

    Google Scholar 

  17. [17] Fuchs, L. Über lineare Differentialgleichungen, welche von Parametern unabhängige Substitutionsgruppen besitzen. Sitzungsberichte der K. preuss. Akademie der Wissenschaften zu Berlin. Einleitung und No. 1–4, 1893, S. 975988; No. 5–8, 1894„ S. 1117–1127. Reprinted in: [FuL], S. 169–195.

    Google Scholar 

  18. [18] Fuchs, L. Über die Abhängigkeit der Lösungen einer linearen differentialgleichung von den in den Coefficienten auftretenden Parametren. Sitzungsberichte der K. preuss. Akademie der Wissenschaften zu Berlin. 1895, S. 905–920. Reprinted in: [FuL], S. 201–217.

    Google Scholar 

  19. Fuchs, R. Sur quelquea équations différentielles linéares du second ordre Compt. Rend. de l’Académie des Sciences, Paris. 141 (1905), pp. 555–558.

    Google Scholar 

  20. ] Fahtmaxep, 4). P. Teopusr.mampuii. 2-e u3darnue. Hayxa, 1966, 575 c. (In Russian). English transl.: GANTMACHER, F.R.. THE THEORY OF MATRICES. Chelsea, New York, 1959, 1960.

    Google Scholar 

  21. ] Garnier, R. Sur une classe d’équations différentielles dont les intégrales générales ont leurs points critiques fixes. Compt. Rend. de l’Académie des Sciences, Paris. 151 (1910), pp. 205–208.

    Google Scholar 

  22. ] Gohberg, I., M.A. Kaashoek, L. Lehrer and L. Rodman. Minimal divisors of rational matrix functions with prescribed zero and pole structures, pp. 241–275 in: Topics in Operator Theory, Systems and Networks. (DYM, H. and I. GOHBERG- ed.) Operator Theory: Advances and Applications, OT 12, Birkhäuser, Basel Boston Stuttgart, 1984.

    Google Scholar 

  23. Harnad, J., Dual isomonodromic tau functions and determinants of inte-grable Fredholm operators. In: Random Matrix Models and Their Applications, (Mathematical Sciences Research Institute Publications, 40), Bleher, P. and A. Its - editors. Cambridge Univ. Press 2001.

    Google Scholar 

  24. ] Harnad, J. and A. ITS. Integrable Fredholm operators and dual isomon-odromic deformations. Commun. Math. Phys., 226 (2002), pp. 497–530.

    Article  MathSciNet  MATH  Google Scholar 

  25. [25] Hartman, PH. Ordinary Differential Equations. Wiley, New York•London•Syd-ney, 1964. Russian transl.: Xaptmah, ‘I. O6bocruoeennbie,flu0Wepeur uacrbnbie ypaenenusr. Mocxsa, Mini), 1970, 720 c.

    Google Scholar 

  26. [26] Hsieh, P., and Y. Sibiya. Basic Theory of Ordinary Differential Equations. Springer-Verlag, New York•Berlin•Heidelberg, 1999, xi+468 pp.

    Google Scholar 

  27. [27] Hukuhara, M. Ordinary Differential Equations. (In Japanese). Iwanami- Zensho 116, Iwanami-Shoten, 1950.

    Google Scholar 

  28. ] Iwasaki, K., Kimura, H., Shimomura, SH. and M. Yoshida. From Gauss to Painlevé. A Modern Theory of Special Functions. (Aspects of mathematics: E, Vol. 16). Vieweg, Braunschweig, 1991.

    Book  Google Scholar 

  29. Jimbo, M., Miwa, T., Môri, Y. and M. Sato. Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica 1D (1980), pp. 80–158.

    Google Scholar 

  30. Jimbo, M., Miwa, T. and K. Ueno. XXXMonodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and T-function. Physica 2D (1981), pp. 306–352.

    MathSciNet  Google Scholar 

  31. M. and T Miwa. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica 2D, (1981), pp. 407–448.

    MathSciNet  Google Scholar 

  32. M. and T Miwa. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III. Physica 4D, (1981), pp. 2646.

    Google Scholar 

  33. Katsnelson, V. Fuchsian differential systems related to rational matrix fuctions in general position and the joint system realization, pp. 117–143 in: Israel Mathematical Conference Proceedings, Vol. 11 (1997), Proceedings of the Ashkelon Workshop on Complex Function Theory (May 1996), Zalcman, L. — editor.

    Google Scholar 

  34. Katsnelson, V. Right and left joint system representation of a rational matrix function in general position (System representation theory for dummies), pp. 337–400 in: Operator theory, system theory and related topics. (The Moshe Livsic anniversary volume. Proceedings of the Conference on Operator Theory held at Ben-Gurion University of the Negev, Beer-Sheva and in Rehovot, June 29—July 4, 1997), ALPAY, D. and V. VINNIKOV - editors. (Operator Theory: Advances and Applications, vol. 123), Birkhäuser Verlag, Basel, 2001.

    Google Scholar 

  35. [35] Kitaev, A.V. and D.A. Korotkin. On solution of the Schlesinger Equations in Terms of 8-functions. Intern. Math. Research Notes, 1998, No. 17, pp. 877905.

    Google Scholar 

  36. [36] I<Opotki4h,.LI. A. H B. B. Matbeeb. O mama-/lyflEt4uona.aanbtx peucenuarx cucmembt ILiaeauuzepa u ypaenenusr Spncma. I yHI<LHOaanbxLH ana.nn3 H ero npn.noz<eam.$, 34:4 (2000), c. 18–34 (Russian). English transi.: KOROTKIN, D.A. and V.B. MATVEEV. On theta function solutions of the Schlesinger system and the Ernst equation. Funk. Anal. Appl., 34:4, (2000), pp. 252–264.

    Google Scholar 

  37. Mahoux, G. Introduction to the theory of isomonodromic deformations of linear ordinary differential equations with rational coefficients. Pp. 35–76 in: The Painlevé Property. One Century Later. CONTE R. — editor. (CRM Series in Mathematical Physics.) Springer-Verlag, New York•Berlin•Heidelberg, 1999.

    Google Scholar 

  38. ] Mason, S.J., M.A. Singer and N.M.J. Woodhouse. Tau function and the twistor theory of integrable systems. Journal of Geometry and Physics, 32 (2000), pp.397–430.

    Article  Google Scholar 

  39. Miwa, T. Painlevé property of monodromy preserving deformation equations and analyticity of r functions. Publ. RIMS Kyoto Univ., 17 (1981), pp. 703–721.

    Article  MathSciNet  MATH  Google Scholar 

  40. [40] Miwa, T., M. JIMBO, and E. DATE. Solitons. Differential equations, symmetries and infinite-dimensional algebras. (Cambridge Tracts in Mathematics, 135.) Cambridge University Press, Cambridge, 2000. x+108 pp.

    Google Scholar 

  41. [41] Newell, A.C. Solitons in Mathematics and Physics. (CBMS-NSF Regional Conference Series in Applied Mathematics, 48.) SIAM, Philadelphia, PA, 1985. xvi+244 pp. Russian transi.: HbIoann, A. Co.iumoubl e MamemamuEe u Ou3toce. MHp, 1989, 326 cc.

    Google Scholar 

  42. ] Palmer, J. Determinants of Cauchy-Riemann operators as r-functions. Acta Applicandae Math, 18 (1990), pp. 199–223.

    Article  Google Scholar 

  43. Palmer, J. Deformation analysis of matrix models. Physica 98D (1994), pp. 166–185.

    MathSciNet  Google Scholar 

  44. Plemelj, J. Riemannsche Funktionenscharen mit gegebener Monodromiegruppe. (Riemann families with prescribed monodromy group). Monatshefte für Math. und Phys., XIX, (1908), pp. 211–245.

    Google Scholar 

  45. Plemelj, J. Problems in the Sense of Riemann and Klein. Intersience Publishers. A division of J. Wiley & Sons Inc., New York • London • Sidney, 1964, 175 pp.

    Google Scholar 

  46. Rasch, G. Zur Theorie und Anwendung des Productintegrales. Journ. für die reine und angew. Math., 171 (1934), pp.65–119.

    Google Scholar 

  47. ] Caxhobiqli, JI.A. O /asmopuaauuu nepedamounoú onepamop-Oysociyuu. IIoxaa,gbI AH CCCP, 226:4 (1976), c. 781–784. Engl. transi.: Sakhnovich, L.A., On the factorization of an operator-valued transfer function. Soviet. Math. Dokl. 17 (1976), pp. 203–207.

    Google Scholar 

  48. Sato, M. The KP hierarchy and infinite-dimensional Grassman manifolds. Proc. of Symposia in Pure Math., 49:1 (1989), pp. 51–66.

    Article  Google Scholar 

  49. Sato, M., Miwa, T and M. Jimbo. Aspects of holonomic quantum fields. Isomonodromic deformations and Ising model. Pp. 429–491 in Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory. Proceeding of the Colloquium held at Les Houches, Centre de Physique, September 1979, Iagolnitzer, D. - editor. (Lectures Notes in Physics, Vol. 126). Springer-Verlag, Berlin•Heidelberg•New York, 1980.

    Google Scholar 

  50. Sato, M., Miwa, T and M. Jimbo. Holonomic quantum fields. II. Publ. RIMS Kyoto Univ. 15 (1979), pp. 201–278.

    Article  MathSciNet  MATH  Google Scholar 

  51. Sato M. and Y. Sato. Soliton equations as dynamical systems on infinite-dimensional Grassmanin manifolds. Pp. 259–271 in: Nonlinear Partial Differential Equations in Applied Science; Proceedings of The U.S.-Japan Seminar, Tokyo, 1982, (Lect. Notes in Num. Appl. Anal., 5), 1982, Fujita, H., P. Lax, and G. Strang — eds.

    Google Scholar 

  52. ] Schlesinger, L. Über die Lösungen gewisser linearer Differentialgleichungen als Funktionen der singulären Punkte. Journal für reine und angew. Math, 129 (1905), pp. 287–294.

    Google Scholar 

  53. ] Schlesinger, L. Vorlesungen über lineare Differentialgleichungen. Leipzig und Berlin, 1908.

    Google Scholar 

  54. ] Schlesinger, L. Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten. Journal für reine und angew. Math, 141 (1912), pp. 96–145.

    MATH  Google Scholar 

  55. [55] Segal, G. and G. Wilson. Loop groups and equations of KdV type. Publ. Mathem. IHES, N° 61 (1985), pp. 5–65. Reprinted in [TeUh], pp. 403–466. Russian transl.: CLn’AJI, r. u,ZJ?K. BMJIbCOH. I’pynna nemeilb u ypaenenusr muna K00. Grp. 379–442 B xaxre. IIPECGJII4, 9. H r. I’pynnba Ilemeab. Mocxsa, Mup, 1990.

    Google Scholar 

  56. [56] Sibuya, Y. Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation. (Translations of Math. Monographs, 82), Amer. Math. Soc., Providence, Rhode Island, 1990, iv+267.

    Google Scholar 

  57. ] Terng, CH. L. and K. Uhlenbeck — editors. Surveys in Differential Geometry: Integral Systems. (Lectures in Geometry and Topology, sponsored by Lehigh University’s Journal of Differential Geometry.) International Press, Boston, 1998.

    Google Scholar 

  58. Wu, T.T., B. Mccoy, C. Tracy and E. Barouch. Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region. Phys. Rev. B, 13:1 (1976), 316–374.

    Google Scholar 

  59. axapob, B.E. H A.B. Iiiabat. Toanasr meopusr dey.aepnoú canto fiosycupoe-scu ‘a odno.’aepnoú.Modyrornuu eo.an a neituneúnou cpede. 9ncnepnM. Teop. 1ns., 61:1, (1971), 118–134 (In Russian). English transl.: Zakharov, V.E. and A.B. SHABAT (=A.B. SABAT). Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media. Soviet Physics JETP, 34:1 (1972), pp. 62–69.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, the Weizmann Institute of Science, Rehovot, 76100, Israel

    V. Katsnelson

  2. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    D. Volok

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  1. V. Katsnelson
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  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Joseph A. Ball  & Martin Klaus  & 

  2. Department of Mathematics, University of California, La Jolla, CA, 92093, USA

    J. William Helton

  3. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Leiba Rodman

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Katsnelson, V., Volok, D. (2004). Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I. In: Ball, J.A., Helton, J.W., Klaus, M., Rodman, L. (eds) Current Trends in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 149. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7881-4_13

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