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Geometric study of Painleve equations and infinite integrable systems

Research Project

Project/Area Number 21740123
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Global analysis
Research InstitutionHitotsubashi University (2011-2012)
Kyushu University (2009-2010)

Principal Investigator

TSUDA Teruhisa  一橋大学, 大学院・経済学研究科, 准教授 (00452730)

Project Period (FY) 2009 – 2012
Project Status Completed (Fiscal Year 2012)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywordsパンルヴェ方程式 / 可積分系 / 超幾何函数
Research Abstract

The UC hierarchy is an extension of the KP hierarchy, which is an infinite-dimensional integrable system associated with the universal rational character of the general linear group. Through a similarity reduction we derive from the UC hierarchy a class of the Schlesinger systems including the Garnier system and the sixth Painleve equation, which describes the monodromy preserving deformations of Fuchsian linear differential equations with certain spectral types. We also present a unified formulation of the above Schlesinger systems as a polynomial Hamiltonian system and their particular solutions in terms of a certain generalization of Gauss' hypergeometric function.

Report

(5 results)
  • 2012 Annual Research Report   Final Research Report ( PDF )
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (32 results)

All 2013 2012 2011 2010 2009 Other

All Journal Article (21 results) (of which Peer Reviewed: 20 results) Presentation (11 results)

  • [Journal Article] UC Hierarchy and Monodoromy Preserving Deformation2013

    • Author(s)
      Teruhisa Tsuda
    • Journal Title

      Journal fur die reine und angewandte Mathematik

      Volume: not yet fixed

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hypergeometric solution of a certain polynomial Hamiltonian system of isomonodromy type2012

    • Author(s)
      T.Tsuda
    • Journal Title

      Quart. J. Math.

      Volume: 63 Pages: 489-505

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] From KP/UC hierarchies to Painleve equations2012

    • Author(s)
      T.Tsuda
    • Journal Title

      Int. J. Math.

      Volume: 23

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Hypergeometric Solution of a Certain Polynomial Hamiltonian System of Isomonodromy Type2012

    • Author(s)
      Teruhisa Tsuda
    • Journal Title

      The Quarterly Journal of Mathematics

      Volume: 63 Issue: 2 Pages: 489-505

    • DOI

      10.1093/qmath/haq040

    • Related Report
      2012 Annual Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] From KP/UC Hierarchy to Painleve Equations2012

    • Author(s)
      Teruhisa Tsuda
    • Journal Title

      International Journal of Mathematics

      Volume: 23

    • Related Report
      2012 Annual Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] UC Hierarchy and Monodoromy Preserving Deformation2012

    • Author(s)
      Teruhisa Tsuda
    • Journal Title

      Journal fur die reine und angewandte Mathematik

      Volume: (印刷中)(掲載確定)

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] N. Nakazono and T.Tsuda,Projective reduction of the discrete Painleve system of type A2 + A12011

    • Author(s)
      K. Kajiwara
    • Journal Title

      Int. Math.Res. Not.

      Volume: 2011 Pages: 930-966

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Projective Reduction of the Discrete Painlev'e System of Type A2+A12011

    • Author(s)
      Kenji Kajiwara, Nobutaka Nakazono, Teruhisa Tsuda
    • Journal Title

      International Mathematics Research Notices

      Volume: Vol.2011(No.4) Pages: 930-966

    • DOI

      10.1093/imrn/rnq089

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] UC階層とモノドロミー保存変形、超幾何函数2011

    • Author(s)
      津田照久
    • Journal Title

      数理解析研究所講究録

      Volume: Vol.1765 Pages: 154-167

    • Related Report
      2011 Annual Research Report
  • [Journal Article] On an integrable system of q-difference equations satisfied by the universal characters: its Lax formalism and an application to q-Painleve equations2010

    • Author(s)
      T.Tsuda
    • Journal Title

      Comm. Math. Phys.

      Volume: 293 Pages: 347-359

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] On an Integrable System of q-Difference Equations Satisfied by the Universal Characters : Its Lax Formalism and an Application to q-Painleve Equations2010

    • Author(s)
      Teruhisa Tsuda
    • Journal Title

      Communications in Mathematical Physics

      Volume: 293 Pages: 347-359

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Projective Reduction of the Discrete Painleve System of Type (A_2+A_1)^<(1)>2010

    • Author(s)
      Kenji Kajiwara, Nobutaka Nakazono, Teruhisa Tsuda
    • Journal Title

      International Mathematics Research Notices

      Volume: (rnq089 first published online)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hypergeometric Solution of a Certain Polynomial Hamiltonian System of Isomonodromy Type2010

    • Author(s)
      Teruhisa Tsuda
    • Journal Title

      The Quarterly Journal of Mathematics

      Volume: (Haq040 first published online)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A. Nobe, H.Tanaka and T.Tsuda, Constructing two-dimensional integrable mappings that possess invariants of high degree2009

    • Author(s)
      J.Matsukidaira
    • Journal Title

      RIMS Koukyuroku Bessatsu

      Volume: B13 Pages: 75-84

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] A. Nobe and T.Tsuda, Ultradiscretization of a solvable two-dimensional chaotic map associated with the Hesse cubic curve2009

    • Author(s)
      K.Kajiwara, M.Kaneko
    • Journal Title

      Kyushu J. Math.

      Volume: 63 Pages: 315-338

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Tropical representation of Weyl groups associated with certain rational varieties2009

    • Author(s)
      T.Tsuda and T.Takenawa
    • Journal Title

      Adv.Math

      Volume: 221 Pages: 936-954

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Universal character and q-Painleve equations2009

    • Author(s)
      T.Tsuda
    • Journal Title

      Math.Ann.

      Volume: 345 Pages: 395-415

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Universal character and g-Painleue equations2009

    • Author(s)
      T.Tsuda
    • Journal Title

      Mathematische Annalen 345

      Pages: 395-415

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Tropical representation of Weyl groups associated with certain rational varieties2009

    • Author(s)
      T.Tsuda, T.Takenawa
    • Journal Title

      Advances in Mathematics 221

      Pages: 936-954

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ultradiscretisation of a solvable two dimersional chaotic map associated with the Hesse cubic curve2009

    • Author(s)
      K.Kajiwara, M.Kaneko, A.Nobe, T.Tsuda
    • Journal Title

      Kyushu Journal of Mathematics 63

      Pages: 315-338

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] UC hierarchy and monodromy preserving deformation

    • Author(s)
      T.Tsuda
    • Journal Title

      J.reine angew. Math.

      Volume: (in press)

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Presentation] UC階層とモノドロミー保存変形,超幾何函数2012

    • Author(s)
      津田照久
    • Organizer
      日本数学会年会(函数方程式論分科会特別講演)
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-26
    • Related Report
      2012 Final Research Report
  • [Presentation] UC階層とモノドロミー保存変形、超幾何函数2012

    • Author(s)
      津田照久
    • Organizer
      日本数学会・関数方程式分科会
    • Place of Presentation
      東京理科大学(特別講演)
    • Year and Date
      2012-03-26
    • Related Report
      2011 Annual Research Report
  • [Presentation] UC hierarchy, monodromy preserving deformation and hypergeometric function2011

    • Author(s)
      T.Tsuda
    • Organizer
      "MPI-Oberseminar" Max Planck Institute for Mathematics
    • Place of Presentation
      Bonn (Germany)
    • Year and Date
      2011-08-25
    • Related Report
      2012 Final Research Report
  • [Presentation] UC hierarchy and monodromy preserving deformations2010

    • Author(s)
      T.Tsuda
    • Organizer
      7th International Conference on "Differential Equations and Dynamical Systems
    • Place of Presentation
      University of South Florida, Tampa (USA)
    • Year and Date
      2010-12-17
    • Related Report
      2012 Final Research Report
  • [Presentation] UC階層とモノドロミー保存変形,超幾何函数2010

    • Author(s)
      津田照久
    • Organizer
      研究集会「可積分系数理の多様性」
    • Place of Presentation
      九州大学応用力学研究所
    • Year and Date
      2010-10-28
    • Related Report
      2012 Final Research Report
  • [Presentation] UC階層とモノドロミー保存変形,超幾何函数2010

    • Author(s)
      津田照久
    • Organizer
      研究集会「可積分系数理の多様性」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2010-08-20
    • Related Report
      2012 Final Research Report
  • [Presentation] UC hierarchy, monodromy preserving deformation and hypergeometric function2010

    • Author(s)
      T.Tsuda
    • Organizer
      International Conference "Symmetry plus Integrability"
    • Place of Presentation
      South Padre Island, Texas (USA)
    • Year and Date
      2010-06-11
    • Related Report
      2012 Final Research Report
  • [Presentation] UC hierarchy, monodoromy preserving deformation and hypergeometric function2010

    • Author(s)
      Teruhisa Tsuda
    • Organizer
      International Conference "Symmetry plus Interability"
    • Place of Presentation
      South Padre Island, Texas (USA)
    • Year and Date
      2010-06-11
    • Related Report
      2010 Annual Research Report
  • [Presentation] UC hierarchy and monodromy preserving deformations2009

    • Author(s)
      T.Tsuda
    • Organizer
      Discrete Integrable Systems
    • Place of Presentation
      Isaac Newton Institute for Mathematical Sciences, Cambridge (UK)
    • Year and Date
      2009-07-03
    • Related Report
      2012 Final Research Report
  • [Presentation] UC hierarchy and monodromy preserving deformations2009

    • Author(s)
      T.Tsuda
    • Organizer
      Discrete Integrable Systems
    • Place of Presentation
      ニュートン研究所(ケンブリッジ大・英国)
    • Year and Date
      2009-07-03
    • Related Report
      2009 Annual Research Report
  • [Presentation] From KP/UC hierarchies to Painleve equations2009

    • Author(s)
      T.Tsuda
    • Organizer
      Discrete Integrable Systems
    • Place of Presentation
      Isaac Newton Institute for Mathematical Sciences,Cambridge (UK)
    • Year and Date
      2009-05-20
    • Related Report
      2012 Final Research Report

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Published: 2009-04-01   Modified: 2019-07-29  

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