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Structural analysis of differential equations by the exact WKB method

Research Project

Project/Area Number 14340042
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

KAWAI Takahiro  KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (20027379)

Co-Investigator(Kenkyū-buntansha) TAKEI Yoshitsugu  KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (00212019)
AOKI Takashi  Kinki University, School of Science and Engineering, Professor, 理工学部, 教授 (80159285)
KOIKE Tatsuya  KYOTO UNIVERSITY, Graduate School of Sciences, Assistant Professor, 大学院理学研究科, 助手 (80324599)
吉野 正史  広島大学, 理学研究科, 教授 (00145658)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥6,600,000 (Direct Cost: ¥6,600,000)
Fiscal Year 2004: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2003: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 2002: ¥2,900,000 (Direct Cost: ¥2,900,000)
Keywordsexact WKB analysis / a higher order Painleve equation / The Toulouse Project / a microdifferential operator of WKB type / a virtual turning point / the exact steepest descent method / singular perturbation theory / algebraic analysis / 高階パンルヴェ方式 / Toulouse Project / 野海・山田系 / Lax対 / ストークス曲線 / ストークス図形 / WKB型擬微分作用素 / 高階バンルヴェ方程式 / ストーク図形 / 西川現象 / 0-パラメタ解 / 第1種の変わり点 / 第2種の変わり点 / 無限個の相 / WKB型微分作用素 / Stokes図形 / 超局所微分作用素 / Landau-Zener型 / 陪特性グラフ
Research Abstract

1.A structure theorem for solutions of a higher order Painleve equation.
Kawai and Takei(Proc. Japan Acad., 80A(2004) have succeeded in transforming any 0-parameter solution of any member of the Painleve hierarchy(P_J)(J=I, II-1,II-2) to that of the classical Painleve-I equation near its turning point of the first kind. This success has made us convinced that the exact WKB analysis should be efficient in studying the structure of higher order Painleve equations, leading to our Announcement of the Toulouse Project(RIMS Kokyuroku 1397(2004)).
2.Proposal of the exact steepest descent method, and its applications. We have confirmed the efficiency of the newly proposed method, the exact steepest descent method, in the situation where the traditional WKB method cannot be applied. (Aoki-Kawai-Takei : Adv. Studies in Pure Math., 42(2004) ; Aoki-Koike-Takei, and Koike-Takei : in Microlocal Analysis and Complex Fourier Analysis, World Scientific,2002).
3.Motivated by a basic integral equation in the plasma physics, we have developed the exact WKB analysis for (micro)differential operators of WKB type. (Aoki-Kawai-Koike-Takei : Adv.in Math.,181(2004),Ann. Inst. Fourier,54 (2004), Contemporary Math.,373(2005), AMS).
4.We have made it manifest that the notion of virtual turning points introduced by Aoki-Kawai-Takei is an indispensable one in the WKB analysis of higher order differential equations. (Kawai-Koike-Nishikawa-Takei : Asterisque,297(2004), Soc. Math. France ; Aoki-Kawai-Sasaki-Shudo-Takei : J.Phys., A38(2005)).

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (42 results)

All 2005 2004 2003 2002 Other

All Journal Article (30 results) Publications (12 results)

  • [Journal Article] On global aspects of exact WKB analysis of operators admitting infinitely many phases.2005

    • Author(s)
      T.Aoki
    • Journal Title

      Contemporary Math. No.373

      Pages: 11-45

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations.2005

    • Author(s)
      T.Aoki
    • Journal Title

      J. Phys. A38

      Pages: 3317-3336

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On global aspects of exact WKB analysis of operators admitting infinitely many phases.2005

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Contemporary Math. 373

      Pages: 11-45

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations.2005

    • Author(s)
      T.Aoki et al.
    • Journal Title

      J.Phys. A38

      Pages: 3317-3336

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations.2005

    • Author(s)
      T.Aoki et al.
    • Journal Title

      J.PHys., A. (in press)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On global aspects of exact WKB analysis of operators admitting infinitely many phases.2005

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Contemporary Math. (in press)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the exact WKB analysis of microdifferential operators of WKB type.2005

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Ann.Inst.Fourier (in press)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the Stokes geometry of higher order Painleve equations.2005

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Asterisque (in press)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the exact WKB analysis of operators admitting infinitely many phases.2004

    • Author(s)
      T.Aoki
    • Journal Title

      Adv. in Math. 181

      Pages: 165-189

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The exact steepest descent method - A new steepest descent method based on the exact WKB analysis.2004

    • Author(s)
      T.Aoki
    • Journal Title

      Adv. Studies in Pure Math. 42

      Pages: 45-61

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the exact WKB analysis of microdifferential operators on WKB type.2004

    • Author(s)
      T.Aoki
    • Journal Title

      Ann. Inst. Fourier 54

      Pages: 1393-1421

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Stokes geometry of higher order Painleve equations.2004

    • Author(s)
      T.Kawai
    • Journal Title

      Asterisque 297

      Pages: 117-166

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On WKB analysis of higher order Painleve equations with a large parameter.2004

    • Author(s)
      T.Kawai
    • Journal Title

      Proc. Japan Acad., Ser. A. 80

      Pages: 53-56

    • NAID

      40006480342

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Announcement of the Toulouse Project, Part22004

    • Author(s)
      T.Kawai
    • Journal Title

      数理解析研究所講究録 1397

      Pages: 172-174

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the exact WKB analysis of operators admitting infinitely many phases.2004

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Adv.in Math. 181

      Pages: 165-189

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The exact steepest descent method- A new steepest descent method based on the exact WKB analysis.2004

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Adv.Studies in Pure Math. 42

      Pages: 45-61

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the exact WKB analysis of microdifferential operators of WKB type.2004

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Ann.Inst.Fourier 54

      Pages: 1393-1421

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Stokes geometry of higher order Painleve equations.2004

    • Author(s)
      T.Kawai et al.
    • Journal Title

      Asterisque 297

      Pages: 117-166

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On WKB analysis of higher order Painleve equations with a large parameter.2004

    • Author(s)
      T.Kawai, Y.Takei
    • Journal Title

      Proc.Japan Acad., Ser.A. 80

      Pages: 53-56

    • NAID

      40006480342

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Announcement of the Toulouse Project, Part22004

    • Author(s)
      T.Kawai, Y.Takei
    • Journal Title

      RIMS Kokyuroku 1397

      Pages: 172-174

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the exact WKB analysis of operators admitting infinitely many phases2004

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Adv.in Math. 181

      Pages: 165-189

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The exact steepest descent method - A new steepest descent method based on the exact WKB analysis2004

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Advanced Studies in Pure Math. 42

      Pages: 45-61

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On WKB analysis of higher order Painleve equations with a large parameter.2004

    • Author(s)
      T.Kawai, Y.Takei
    • Journal Title

      Proc.Japan Acad.Ser.A 80

      Pages: 53-56

    • NAID

      40006480342

    • Related Report
      2004 Annual Research Report
  • [Journal Article] WKB analysis of higher order Painleve equations with a large parameter - Local reduction of 0-parameter solutions for Painleve hierarchies (P_J)(J=I,II-1 or II-2).2004

    • Author(s)
      T.Kawai, Y.Takei
    • Journal Title

      RIMS preprint 1468

      Pages: 1-36

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 完全WKB解析,そして完全最急降下法-特異摂動の代数解析学続論2003

    • Author(s)
      Y.Takei
    • Journal Title

      数学 55

      Pages: 350-367

    • NAID

      10013701687

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Exact WKB analysis and the exact steepest descent method-A sequel to "Algebraic analysis of singular parturbations" (in Japanese)2003

    • Author(s)
      Y.Takei
    • Journal Title

      Sugaku 55

      Pages: 350-367

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Vanishing of Stokes curves.2002

    • Author(s)
      T.Aoki
    • Journal Title

      Microlocal Analysis and Complex Fourier Analysis

      Pages: 1-22

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The effect of new Stokes curves in the exact steepest descent method.2002

    • Author(s)
      T.Koike
    • Journal Title

      Microlocal Analysis and Complex Fourier Analysis

      Pages: 186-199

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Vanishing of Stokes curves.2002

    • Author(s)
      T.Aoki et al.
    • Journal Title

      Microlocal Analysis and Complex Fourier Analysis, World Scientific

      Pages: 1-22

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The effect of new Stokes curves in the exact steepest descent method.2002

    • Author(s)
      T.Koike, Y.Takei
    • Journal Title

      Microlocal Analysis and Complex Fourier Analysis, World Scientific

      Pages: 186-199

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Takahiro Kawai, T.Aoki, T.Koike, Y.Takei: "On the exact WKB analysis of operators admitting infinitely many phases"Adv. in Math.. 181. 165-189 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takahiro Kawai, T.Aoki, Y.Takei: "The exact steepest descent method - A new steepest descent method based on the exact WKB analysis"Adv. in Pure Math.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] Takahiro Kawai, T.Aoki, T.Koike, Y.Takei: "On global aspects of exact WKB analysis of operators admitting infinitely many phases"Contemporary Math.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] Takahiro Kawai, T.Aoki, T.Koike, Y.Takei: "On the exact WKB analysis of microdifferential operators of WKB type"RIMS preprint 1429. (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takahiro Kawai, T.Koike, Y.Nishikawa, Y.Takei: "On the Stokes geometry of higher order Painleve equations"RIMS preprint 1443. (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] [AKKT1]T.Aoki, T.Kawai, T.Koike, Y.Takei: "On global aspects of exact WKB analysis of operators admitting infinitely many phases"RIMS Preprint 1392. (発表予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] [AKKT2]T.Aoki, T.Kawai, T.Koike, Y.Takei: "On the exact WKB analysis of operators admitting infinitely many phases"Adv.in Math.に近刊. (発表予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] [AKT1]T.Aoki, T.Kawai, Y.Takei: "Exact WKB analysis of non-adiabatic transition probabilities for three levels"J.Phys.. A35. 2401-2430 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] [AKoT]T.Aoki, T.Koike, Y.Takei: "Vanishing of Stokes curves"Microlocal Analysis and Complex Fourier Analysis, World Scientific. 1-22 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] [KS] T.Kawai, H.P.Stapp: "On infra-red singularities associated with QC photons"Microlocal Analysis and Complex Fourier Analysis, World Scientific. 115-134 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] [KoT] T.Koike, Y.Takei: "The effect of new Stokes curves in the exact steepest descent method"Microlocal Analysis and Complex Fourier Analysis, World Scientific. 186-199 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] [AKY]青木貴史, 片岡清臣, 山崎晋: "超函数・FBI変換・無限階擬微分作用素"共立出版 近刊. (2003)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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