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Global Analysis of Painleve Equations

Research Project

Project/Area Number 10440058
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKobe University

Principal Investigator

TAKANO Kyoichi  Kobe Univ Faculty of Science, Professor, 理学部, 教授 (10011678)

Co-Investigator(Kenkyū-buntansha) YAMADA Yasuhiko  Kobe Univ. Grad. School of S.& T., Asso. Prof, 自然科学研究科, 助教授 (00202383)
NOUMI Masatoshi  Kobe Univ. Grad. School of S.& T., Professor, 自然科学研究科, 教授 (80164672)
SASAKI Takeshi  Kobe Univ. Faculty of Science, Professor, 理学部, 教授 (00022682)
TAKEI Yoshitsugu  Kyoto Univ. RIMS, Asso. Prof., 数理解析研究所, 助教授 (00212019)
IWASAKI Katsunori  Kyushu Univ. Dept. of Math., Professor, 数理学研究科, 教授 (00176538)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥9,200,000 (Direct Cost: ¥9,200,000)
Fiscal Year 2000: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1999: ¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 1998: ¥3,800,000 (Direct Cost: ¥3,800,000)
KeywordsPainleve equations / Higher order Painleve equations / Garnier systems / Affine Weyl group symmetries / Backlund transformations / Spaces of initial conditions / Hamiltonian structures / Exact WKB analysis / シュヴァルツ写像 / 逆分岐問題 / ドリンフェルト・ソコロフ階層 / フロベニウス多様体 / 完全WKB法 / WKB法 / バーコフの標準形 / ゲーゲンバウアー多項式
Research Abstract

1. Symmetries of Painleve equations : Theory of Backlund transformations (realization of affine Weyl groups) for Painleve equations has been constructed. The theory gives not only good perspective but also usefull tools to the study of Painleve equations. For example, we can easily obtain the form of each Backlund transformation as birational transformation and various kinds of special polynomials associated with Painleve equations. Similar theory is now being devepoled for discrete Painleve equations.
2. The spaces of initial conditions : (1) A relation between the spaces of initial conditions and Backlund transformations has been made clear, namely, the manifold obtained by patching affine charts via Backlund transformations are isomorphic to Okamoto's space of initial conditions. Fromx this fact, we can derive that the spaces of initial conditions whose papameters are equivalent under the affine Weyl group are isomorphic to each other. (2) Spaces of initial conditions for a higher order Painleve equation of type A^<(1)>_4 and degenerated Garnier systems of two variables have been obtained.
3. Exact WKB analysisi : (1) The connection problem for the second Painleve equation with a large paraneter has been solved by the use of exact WKB analysis. The connection formulas are given by compositions of those for the first Painleve equation with a large parameter. For this purpose, a reduction theorem to Birkhoff's normal form has been shown. The usual steepest descent method has been extended to one for third order linear differential equations.
4. Hypergeometric equations : A problem of studying Schwarz theory in the case where all parameters are pure imaginary numbers has been proposed. Some experiments were carried out.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (41 results)

All Other

All Publications (41 results)

  • [Publications] K.Takano: "Defining manifolds for Painleve equations"Toward the Exact WKB Analysis of Differential Equations. 261-269 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Sasaki: "Projective surfaces defined by Appell's hypergeometric systems E_4 and E_2"Kyushu Journal of Mathematics. 55. 1-21 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Noumi: "Affine Weyl groups, discrete dynamical systems and Painleve equations"Comm.Math.Phys.. 199. 281-295 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Noumi: "Affine Weyl group symmetries in Painleve type equations"Toward the Exact WKB Analysis of Differential Equations. 245-259 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Iwasaki: "Recent progress in polyhedral harmonics"Acta Applicandae Mathematicae. 60. 179-197 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kawai: "WKB analysis of Painleve transcendents with a large parameter III"Adv.in Math.. 134. 178-218 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 河合隆裕: "特異摂動の代数解析学"岩波書店. 132 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 野海正俊: "パンルヴェ方程式-対称性からの入門"朝倉書店. 204 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Takano: "Difining manifolds for Painleve equations"in "Toward the exact WKB analysis of differential equations, linear or nonlinear"(Eds.T.Kawai and Y.Takei), Kyoto Univ. Press, Kyoto.. 261-269 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Takano: "Confluence processes in defining manifolds for Painleve systems"to appear in. Tohoku J.Math..

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Sasaki: "Projective surfaces defined by Appell's hypergeometric systems E_4 and E_2"Kyushu J.Math.. 55. 1-21 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Noumi and Y.Yamada: "Affine Weyl groups, discrete dynamical systems and Painleve equations"Comm.Math.Phys.. 199. 281-295 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Noumi and Y.Yamada: "Affine Weyl group symmetries in Painleve type equations"in"Toward the exact WKB analysis of differential equations, linear or non-linear"(Eds.C.J.Howls, T.Kawai, Y.Yakei), Kyoto Unviersity Press, Kyoto. 245-259 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Iwasaki: "Polytopes, invariants and harmonic functions"in"Arrangements-Tokyo 1998", M.J.Falk and H.Terao eds., Advanced Studies in Pure Mathematics 27, Kinokuniya, Tokyo. 145-156 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Iwasaki and Y.Kamimura: "Inverse bifurcation problem, singular Wiener-Hopf equations and mathematical models in ecology"Journal of Mathimatical Biology.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kawai, Y.Takei: "WKB analysis of Painleve transcendents with a large parameter."Adv. in Math.. 134. 178-218 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kawai and Y.Takei: "Algebraic Analysis of Singular Perturbations (in Japanese)"Iwamami, Tokyo, Japan. 132 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Eds.K.Takano and M.Noumi: "Prospective of Painleve Equations (in Japanese) Rokko Lectures in Mathematics 7"Kobe Univ. 194 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Noumi: "Painleve Equations-An Introducation from the Viewpoint of Symmetries (in Japanese)"Asakura, Tokyo, Japan. 204 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Takano: "Confluence processes in defining manifolds for Painleve systems"Tohoku Journal of Mathematics. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Sasaki: "Projective surfaces defined by Appell's hypergeometric systems E_4 and E_2"Kyushu Journal of Mathematics. 55. 1-21 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Noumi: "Affine Weyl group symmetries in Painleve type equations"Toward the Exact WKB Analysis of Differential Equations. 245-259 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Kajiwara: "Determinant formulas for the Toda and discrete Toda Equations"Funkcialaj Elvacioj. (to appear). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Iwasaki: "Recent progress in polyhedral harmonics"Acta Applicandae Mathematicae. 60. 179-197 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Aoki: "On the exact steepest descent method -a new method for the description of Stokes curves"Journal of Mathematical Physics. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] 高野恭一: "パンルヴェ方程式の眺望"Department of Mathematics, Kobe University. (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 野海正俊: "パンルヴェ方程式-対称性からの入門"朝倉書店. 204 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Takano: "Defining Manifolds for Painleve Equations"Toward the Exact WKB Analysis of Differential Equations. 261-269 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Matano: "On Some Hamiltonian Structures of Painleve Systems,II"Journal of the Mathematical Society of Japan. 51-4. 843-866 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Noumi: "Symmetries in the Fourth Painleve Equation and Okamoto Polynomials"Nagoya Mathematical Journal. 153. 53-86 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Sasaki: "The Uniformizing Differential Equation of the Complex Hyperbolic Structure on the Moduli Space of Marked Cubic Surfaces"Proceedings of Japan Academy,Ser.A. 75. 129-133 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Iwasaki: "A Combinatorial Formula of Leibniz Type with Application"Proceedings of the American Mathematical Society. 127. 29-33 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Takei: "An Explicit Description of the Connection Formula for the First Painleve Equation"Toward the Exact WKB Analysis of Differential Equations. 271-296 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] C.J.Howls: "Toward the Exact WKB Analysis of Differential Equations,Linear or Non-Linear"京都大学学術出版会. 298 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Matano: "On Some Hamiltonian Strucures of Painleve Systems,II" Journal of the Mathematical Society of Japan. (to appear). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Noumi: "Umemura Polynomials for Painleve V Equation" Physics Letters.A. 247. 65-69 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Noumi: "Higher Order Painleve Equations of Type A_l^<(1)>" Funkcialaj Ekvacioj. 41. 363-381 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Noumi: "Affine Weyl Groups,Discrete Dynamical Systems and Painleve Equations" Communications in Mathematical Physics. 199. 281-295 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Iwasaki: "A Combinatorial Formula of Leibniz Type with Application" Proceedings of the American Mathematical Society. 127. 29-33 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Takei: "Singular-Perturbative Reduction to Birkhoff Normal Form" Publications of the Research Institute for Mathematical Sciences. 34. 601-627 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 河合隆裕: "特異摂動の代数解析学" 岩波書店, 132 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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