Geometric study of Painleve equations and infinite integrable systems

• Project/Area Number: 25870234
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis; Mathematical analysis
• Research Institution: Hitotsubashi University
• Principal Investigator: Tsuda Teruhisa 一橋大学, 大学院経済学研究科, 准教授 (00452730)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: パンルヴェ方程式 / 超幾何函数 / 無限可積分系 / 有理函数近似 / 連分数 / タウ函数

Outline of Final Research Achievements

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painleve; equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.

Research Products

• [Journal Article] Hermite-Pade approximation, isomonodromic deformation and hypergeometric integral (2017)
  • Author(s): Toshiyuki Mano and Teruhisa Tsuda
  • Journal Title: Mathematische Zeitschrift Volume: 285 Issue: 1-2 Pages: 397-431
  • DOI: 10.1007/s00209-016-1713-y
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Hermite-Pade approximation, isomonodromic deformation and hypergeometric integral (2016)
  • Author(s): T. Mano and T. Tsuda
  • Journal Title: Mathematische Zeitschrift Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On a fundamental system of solutions of a certain hypergeometric equation (2015)
  • Author(s): T. Tsuda
  • Journal Title: Ramanujan Journal Volume: 38 Issue: 3 Pages: 597-618
  • DOI: 10.1007/s11139-014-9630-3
  • Peer Reviewed
• [Journal Article] Hermite による２つの近似問題と Schlesinger 変換 (2014)
  • Author(s): 眞野智行、津田照久
  • Journal Title: RiMS 講究録別冊 Volume: B47 Pages: 77-86
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] パンルヴェ方程式と連分数 (2014)
  • Author(s): 津田照久
  • Journal Title: 数学セミナー Volume: 634 Pages: 30-37
• [Journal Article] Hermite による２つの近似問題と Schlesinger 変換 (2014)
  • Author(s): 眞野智行、津田照久
  • Journal Title: RIMS 講究録別冊 Volume: not yet fixed
  • Peer Reviewed
• [Presentation] Hermite-Pade approximation, isomondromic deformation and hypergeometric integral (2016)
  • Author(s): Teruhisa Tsuda
  • Organizer: Workshop ``Painleve Equations and Discrete Dynamics"
  • Place of Presentation: BIRS, Banf, Canada
  • Year and Date: 2016-10-03
  • Int'l Joint Research / Invited
• [Presentation] Painleve equation and continued fraction (2016)
  • Author(s): Teruhisa Tsuda
  • Organizer: Discrete Integrable Systems Workshop
  • Place of Presentation: TSIMF, Sanya, China
  • Year and Date: 2016-04-11
  • Int'l Joint Research / Invited
• [Presentation] Continued fractions and Painleve equations (2015)
  • Author(s): T. Tsuda
  • Organizer: "Differential and Difference Equations: Analytic, Arithmetic and Galoisian Approaches"
  • Place of Presentation: Laboratoire Paul Painleve, Lille University, Lille (France)
  • Year and Date: 2015-10-23
  • Int'l Joint Research / Invited
• [Presentation] Hermite-Pade approximation, isomonodromic deformation and hypergeometric integral (2015)
  • Author(s): T. Tsuda
  • Organizer: The 9th IMACS International Conference on "Nonliner Evolution Equations and Wave Phenomena: Computation and Theory"
  • Place of Presentation: Georgia University, Athens (USA)
  • Year and Date: 2015-04-03
  • Int'l Joint Research / Invited
• [Presentation] Continued fractions and Painleve equations (2014)
  • Author(s): Teruhisa Tsuda
  • Organizer: RIMS Workshop "Recent developments in differential equations in the complex domain"
  • Place of Presentation: 京都大学数理解析研究所（京都府京都市）
  • Year and Date: 2014-11-19
  • Invited
• [Presentation] Hermite-Pade 近似，モノドロミー保存変形，繰り返し超幾何積分
  • Author(s): 津田照久
  • Organizer: RIMS研究集会「非線形離散可積分系の新展開」
  • Place of Presentation: 京都大学数理解析研究所（京都）
  • Invited