Connection problems for Painleve hierarchies and WKB analysis

2008 Fiscal Year Final Research Report

• Project/Area Number: 18540174
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: TAKEI Yoshitsugu Kyoto University, 数理解析研究所, 准教授 (00212019)
• Co-Investigator(Kenkyū-buntansha): KOIKE Tatsuya 神戸大学, 大学院・理学研究科, 准教授 (80324599)
• Co-Investigator(Renkei-kenkyūsha): KOIKE Tatsuya 神戸大学, 大学院・理学研究科, 准教授 (80324599)
• Project Period (FY): 2006 – 2008
• Keywords: 代数解析 / WKB解析 / パンルベ階層 / ガルニエ系

Research Abstract

本研究では、完全WKB解析を用いて高階パンルベ方程式の接続問題を解明することを目的として研究を進め、高階パンルベ方程式の階層に対する完全WKB解析の枠組をほぼ構築することに成功した。具体的には、接続問題を論じる際の主役となる(インスタントン型)形式解の構成、第一種変わり点における局所的構造定理(即ち、第一種変わり点では2階のパンルベI型方程式へ変換できること)の確立、第二種変わり点での標準型の発見、等の諸結果が得られた。

Research Products

• [Journal Article] The Bender-Wu analysis and the Voros theory. II (2009)
  • Author(s): T. Aoki, T. Kawai,and Y. Takei
  • Journal Title: Advanced Studies in Pure Mathematics 54巻 Pages: 19-94
  • Peer Reviewed
• [Journal Article] Sato's conjecture for the Weber equation and transformation theory for Schrodinger equations with a merging pair of turning points (2008)
  • Author(s): Y. Takei
  • Journal Title: RIMS Kokyuroku Bessatsu B10巻 Pages: 205-224
  • Peer Reviewed
• [Journal Article] Half of the Toulouse Project Part 5 is completed - Structure theorem for instanton-type solutions of (PJ)_m (J=I, II or IV) near a simple P-turning point of the first kind (2008)
  • Author(s): T. Kawai,and Y. Takei
  • Journal Title: RIMS Kokyuroku Bessatsu B5巻 Pages: 99-115
  • Peer Reviewed
• [Journal Article] On new expressions of the Painleve hierarchies (2008)
  • Author(s): T. Koike
  • Journal Title: RIMS Kokyuroku Bessatsu B5巻 Pages: 153-198
  • Peer Reviewed
• [Journal Article] Virtual turning points - A gift of microlocal analysis to the exact WKB analysis (2008)
  • Author(s): T. Aoki, N. Honda, T. Kawai, T. Koike, Y. Nishikawa, S. Sasaki, A. Shudo,and Y. Takei
  • Journal Title: Algebraic Analysis of Differential Equations Pages: 29-43
  • Peer Reviewed
• [Journal Article] Instanton-type formal solutions for the first Painleve hierarchy (2008)
  • Author(s): Y. Takei
  • Journal Title: Algebraic Analysis of Differential Equations Pages: 307-319
  • Peer Reviewed
• [Journal Article] Exact WKB analysis, and exact steepest descent method - A sequel to "algebraic analysis for singular perturbations" - (2007)
  • Author(s): Y. Takei
  • Journal Title: Sugaku Expositions 20巻 Pages: 169-189
  • Peer Reviewed
• [Journal Article] Exact WKB analysis for the degenerate third Painleve equation of type (D8) (2007)
  • Author(s): Y. Takei,and H. Wakako
  • Journal Title: Proceedings of the Japan Academy, Series A 83巻 Pages: 63-68
  • Peer Reviewed
• [Journal Article] On the Hamiltonian structures of the second and the fourth Painleve hierarchies, and the degenerate Garnier systems (2007)
  • Author(s): T. Koike
  • Journal Title: RIMS Kokyuroku Bessatsu B2巻 Pages: 99-127
  • Peer Reviewed
• [Journal Article] Toward the exact WKB analysis for instanton-type solutions of Painleve hierarchies (2007)
  • Author(s): Y. Takei
  • Journal Title: RIMS Kokyuroku Bessatsu B2巻 Pages: 247-260
  • Peer Reviewed
• [Journal Article] On a local reduction of a higher order Painleve equation and its underlying Lax pair near a simple turning point of the first kind (2006)
  • Author(s): Y. Takei
  • Journal Title: Seminaires et Congres 14巻 Pages: 251-267
  • Peer Reviewed
• [Journal Article] WKB analysis of higher order Painleve equations with a large parameter (2006)
  • Author(s): T. Kawai,and Y. Takei
  • Journal Title: Advances in Mathematics 203巻 Pages: 636-672
  • Peer Reviewed
• [Presentation] Exact WKB analysis for simple-pole type singularities of Painleve equations (2009)
  • Author(s): Y. Takei
  • Organizer: 数理物理学セミナー
  • Place of Presentation: Paris第7大学(France)
  • Year and Date: 2009-03-17
• [Presentation] On the summability and Stokes phenomena for formal solutions of some PDEs - An exact WKB approach (2009)
  • Author(s): Y. Takei
  • Organizer: 微分方程式論セミナー
  • Place of Presentation: Angers大学(France)
  • Year and Date: 2009-03-13
• [Presentation] Transformation theory for MTP equations and singularity structure of the Borel transform of their WKB solutions (2008)
  • Author(s): Y. Takei
  • Organizer: Foundations of exact WKB analysis and resurgence theory
  • Place of Presentation: 関西セミナーハウス
  • Year and Date: 20081217-18
• [Presentation] On the summability and Stokes phenomena for formal solutions of some PDEs - An exact WKB approach (2008)
  • Author(s): Y. Takei
  • Organizer: 解析的な方程式の新しい取り組み
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-09-17
• [Presentation] On the exact WKB analysis for the degenerate third Painleve equation (2008)
  • Author(s): Y. Takei
  • Organizer: Exact WKB Analysis and Microlocal Analysis
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-05-27
• [Presentation] On the transformation theory for Schrodinger equations with two merging turning points (2008)
  • Author(s): Y. Takei
  • Organizer: 偏微分方程式姫路研究集会
  • Place of Presentation: 兵庫県立大学
  • Year and Date: 2008-02-22
• [Presentation] On the eigenvalue problem of the Coulomb potential and exact WKB analysis (2008)
  • Author(s): Y. Takei
  • Organizer: Holomorphic partial differential equations, small divisors and summability
  • Place of Presentation: Centre International de Rencontres Mathematiques(Marseille, France)
  • Year and Date: 2008-01-29
• [Presentation] 単純極型作用素の完全WKB解析について (2008)
  • Author(s): Y. Takei
  • Organizer: 超幾何方程式研究会2008
  • Place of Presentation: 神戸大学
  • Year and Date: 2008-01-07
• [Presentation] The Bender-Wu analysis and the Voros theory (Part 2) (2007)
  • Author(s): T. Aoki, T. Kawai,and Y. Takei
  • Organizer: Differential Equations and Exact WKB Analysis
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 20071011-12
• [Presentation] On the theory of transformation for operators with two merging simple turning points (2007)
  • Author(s): Y. Takei
  • Organizer: Recent development of global and micro-local analysis on partial differential equations
  • Place of Presentation: 龍谷大学
  • Year and Date: 2007-09-14
• [Presentation] Structure of instanton-type solutions of higher order Painleve equations near a simple turning point of the first kind (2006)
  • Author(s): T. Kawai,and Y. Takei
  • Organizer: Algebraic Analysis and the Exact WKB Analysis for Systems of Differential Equations
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2006-12-12
• [Presentation] Toward the exact WKB analysis for instanton-type solutions for Painleve hierarchies (2006)
  • Author(s): Y. Takei
  • Organizer: Workshop on Applied Asymptotics and Modelling
  • Place of Presentation: International Centre for Mathematical Sciences(Edinburgh, UK)
  • Year and Date: 2006-06-30
• [Presentation] Toward the exact WKB analysis for instanton-type solutions of Painleve hierarchies (2006)
  • Author(s): Y. Takei
  • Organizer: Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and their Deformations. Painleve Hierarchies
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2006-05-20
• [Book] 特異摂動の代数解析学(1998年発行の岩波講座版の単行本化) (2008)
  • Author(s): 河合隆裕, 竹井義次
  • Total Pages: 132
  • Publisher: 岩波書店
• [Book] Springer-Verlag (2008)
  • Author(s): T. Aoki, H. Majima, Y. Takei,and N
  • Total Pages: 352
  • Publisher: Algebraic Analysis of Differential Equations - from Microlocal Analysis to Exponential Asymptotics, Festschrift in Honor of Takahiro Kawai