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Theory of integrable hierarchies and its application to mathematical physics

Research Project

Project/Area Number 22540186
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TAKASAKI Kanehisa  京都大学, 大学院・人間環境学研究科, 教授 (40171433)

Co-Investigator(Renkei-kenkyūsha) SASAKI Ryu  京都大学, 基礎物理学研究所, 准教授 (20154007)
Project Period (FY) 2010 – 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: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2010: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords代数解析 / 位相的弦理論 / 一般化コニフォールド / q差分方程式 / 量子ミラー曲線 / 溶解結晶模型 / 量子トーラス代数 / 戸田階層 / アブロビッツ・ラディック階層 / リーマン・ヒルベルト問題 / アブロヴィッツ・ラディック階層 / 無分散極限 / フロベニウス構造 / 準古典展開 / BD型可積分階層 / 一般化弦方程式 / フルヴィッツ数 / スペクトル曲線 / 一般化ホドグラフ法 / 対数的時間発展
Research Abstract

Integrable hierarchies are a class of classical integrable system. Typical examples are the KP hierarchy, the Toda hierarchy and multi-component versions thereof. This research is focussed on both theoretical aspects and applications to mathematical physics. All materials play an important role in recent progress of mathematical physics. As regards the theoretical aspect, we have obtained some results on the Toda hierarchy, the dispersionless Toda hierarchy and integrable hierarchies of the B and D types. Subjects of applications to mathematical physics range over statistical mechanics, gauge theories, string theories and related mathematical issues. Solvable models in discrete quantum mechanics are also studied in the context of orthogonal polynomials.

Report

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

    (29 results)

All 2013 2012 2011 2010

All Journal Article (12 results) (of which Peer Reviewed: 11 results) Presentation (17 results) (of which Invited: 1 results)

  • [Journal Article] Modified melting crystal model and Ablowitz-Ladik hierarchy2013

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 46

    • NAID

      120005296111

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Old and new reductions of dispersionless Toda hierarchy2012

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      SIGMA 8

      Volume: 102 Pages: 22-22

    • DOI

      10.3842/sigma.2012.102

    • Related Report
      2012 Annual Research Report 2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Multi-indexed (q-)Racah Polynomials2012

    • Author(s)
      Satoru Odake and Ryu Sasaki
    • Journal Title

      Journal of Physics A: Mathematical and Theoremtical

      Volume: 45

    • NAID

      120007114349

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] An hbar-expansion of the Toda hierarchy: a recursive construction of solutions2012

    • Author(s)
      Kanehisa Takasaki and Takashi Takebe
    • Journal Title

      Analysis and Mathematical Physics

      Pages: 171-214

    • Related Report
      2012 Annual Research Report 2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] hermodynamic limit of random partitions and dispersionless Toda hierarchy2012

    • Author(s)
      Kanehisa Takasaki and Toshio Nakatsu
    • Journal Title

      J. Phys. A: Math. Theor.

      Volume: 45

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] An hbar-dependent formulation of the Kadomotsev-Petviashvili hierarchy2012

    • Author(s)
      Kanehsia Takasaki and Takashi Takebe
    • Journal Title

      Theoretical and Mathematical Physics

      Volume: 171 Pages: 683-690

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Generalized string equations for double Hurwitz numbers2012

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Journal of Geometry and Physics

      Volume: 62 Pages: 1135-1156

    • NAID

      120003988367

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] hbar-Dependent KP hierarchy2012

    • Author(s)
      Kanehisa Takasaki and Takashi Takebe
    • Journal Title

      Theoretical and Mathematical Physics

      Volume: 171 (2) Issue: 2 Pages: 683-690

    • DOI

      10.1007/s11232-012-0065-y

    • Related Report
      2012 Annual Research Report
  • [Journal Article] Generalized string equations for double Hurwitz numbers2012

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Journal of Greometry and Physics

      Volume: 62 Issue: 5 Pages: 1135-1156

    • DOI

      10.1016/j.geomphys.2011.12.005

    • NAID

      120003988367

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Thermodynamic limit of random partitions and dispertionless Todahierarchy2012

    • Author(s)
      高崎金久、中津了勇
    • Journal Title

      J. of Phys. A : Math. Theor

      Volume: Vol 45 Issue: 2 Pages: 25403-25403

    • DOI

      10.1088/1751-8113/45/2/025403

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Toda tau functions with quantum torus symmetries2011

    • Author(s)
      Kanehisa Takasaki
    • Journal Title

      Acta Polytechnica

      Volume: 51 Pages: 74-76

    • Related Report
      2012 Final Research Report 2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Discrete quantum mechanics2011

    • Author(s)
      Satoru Odake and Ryu Sasaki
    • Journal Title

      Journal of Physics A: Mathematical and Theoretical

      Volume: 44 Issue: 35 Pages: 353001-353001

    • DOI

      10.1088/1751-8113/44/35/353001

    • NAID

      120007114350

    • Related Report
      2012 Final Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Presentation] 溶解結晶模型とAblowitz-Ladki 階層2013

    • Author(s)
      高崎金久
    • Organizer
      日本数学会年会
    • Place of Presentation
      京都大学
    • Year and Date
      2013-03-22
    • Related Report
      2012 Final Research Report
  • [Presentation] 溶解結晶模型とAblowitz-Ladik階層2013

    • Author(s)
      高崎金久
    • Organizer
      日本数学会年会
    • Place of Presentation
      京都大学
    • Related Report
      2012 Annual Research Report
  • [Presentation] Exactly Solvable Quantum Mechanics and Multi-indexed Orthogonal Polynomials2012

    • Author(s)
      Ryu Sasaki
    • Organizer
      The 29 International Colloquium on Group-Theoretical Methods in Physics
    • Place of Presentation
      Chern Institute of Mathematics, Tianjin, China
    • Year and Date
      2012-08-26
    • Related Report
      2012 Final Research Report
  • [Presentation] Integrable structure of melting crystal model2012

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      Integrability in Gauge and String Theory
    • Place of Presentation
      スイス連邦工科大学チューリッヒ校
    • Year and Date
      2012-08-23
    • Related Report
      2012 Final Research Report
  • [Presentation] Combinatorial properties of toric topological string partition functions2012

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      合宿型セミナー「ヤング図形・統計物理に関連する代数的組合せ論」(招待講演)
    • Place of Presentation
      国際高等研究所 .招待あり
    • Year and Date
      2012-08-09
    • Related Report
      2012 Final Research Report
  • [Presentation] Combinatorial properties of toric topological string partition functions2012

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      合宿型セミナー「ヤング図形・統計物理に関連する代数的組合せ論」
    • Place of Presentation
      国際高等研究所
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Integrable structure of modified melting crystal mode2012

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      Integrability in Gauge and String Theory
    • Place of Presentation
      スイス連邦工科大学チューリッヒ校
    • Related Report
      2012 Annual Research Report
  • [Presentation] Non-degenerate solutions of universal Whitham hierarchy2011

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      7th International Conference on Nonlinear Evolution Equations and Wave Phenomena : Computation and Theory
    • Place of Presentation
      University of Georgia, Athens, USA(招待講演)
    • Year and Date
      2011-04-05
    • Related Report
      2010 Annual Research Report
  • [Presentation] 無分散変形 KP 階層の新しい簡約2011

    • Author(s)
      高崎金久
    • Organizer
      日本数学会年会
    • Place of Presentation
      早稲田大学(備考:東日本大震災によって中止されたが,講演アブストラクト集の発行によって発表は行われたものとみなされている)
    • Year and Date
      2011-03-22
    • Related Report
      2012 Final Research Report
  • [Presentation] 無分散変形KP階層の新しい簡約2011

    • Author(s)
      高崎金久
    • Organizer
      日本数学会年会無限可積分系セッション
    • Place of Presentation
      備考:震災によって年会(早稲田大学)は中止されたが,講演アブストラクト集の発行によって発表は行われたものとみなされる
    • Year and Date
      2011-03-22
    • Related Report
      2010 Annual Research Report
  • [Presentation] Hbar-expansion of integrable hierarchies: recursive construction of solutions2011

    • Author(s)
      Takashi Takebe
    • Organizer
      Iinternational Workshop on Classical and Quantum Integrable Systems
    • Place of Presentation
      nstitute for High Energy Physics, Protvino, Russia.招待あり
    • Year and Date
      2011-01-27
    • Related Report
      2012 Final Research Report
  • [Presentation] Hbar-expansion of integrable hierarchies : recursive construction of solutions2011

    • Author(s)
      Takashi Takebe
    • Organizer
      International Workshop on Classical and Quantum Integrable Systems
    • Place of Presentation
      Institute for High Energy Physics, Protvino, Russia(招待講演)
    • Year and Date
      2011-01-27
    • Related Report
      2010 Annual Research Report
  • [Presentation] フルヴィッツ数に関連する戸田階層の特殊解とその古典極限2010

    • Author(s)
      高崎金久
    • Organizer
      日本数学会秋期総合分科会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-24
    • Related Report
      2012 Final Research Report
  • [Presentation] フルヴィッツ数に関連する戸田階層の特殊解とその古典極限2010

    • Author(s)
      高崎金久
    • Organizer
      日本数学会秋期総合分科会無限可積分系セッションー
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-24
    • Related Report
      2010 Annual Research Report
  • [Presentation] Exactly Solvable (Discrete) Quantum Mechanics and Exceptional Orthogonal Polynomial2010

    • Author(s)
      Ryu Sasaki
    • Organizer
      The Second International Conference: Nonlinear Waves --Theory and Applications
    • Place of Presentation
      Tsinghua University, Beijing, China
    • Year and Date
      2010-06-27
    • Related Report
      2012 Final Research Report
  • [Presentation] Exactly Solvable(Discrete) Quantum Mechanics and Exceptional Orthogonal Polynomials2010

    • Author(s)
      Ryu Sasaki
    • Organizer
      The Second International Conference : Nonlinear Waves--Theory and Applications
    • Place of Presentation
      Tsinghua University, Beijing, China(招待講演)
    • Year and Date
      2010-06-27
    • Related Report
      2010 Annual Research Report
  • [Presentation] Toda tau function with quantum torus symmetries2010

    • Author(s)
      Kanehisa Takasaki
    • Organizer
      19th International Colloquium on Integrable Systems and Quantum Symmetries
    • Place of Presentation
      Czech Technical University, Prague, Czech Republic
    • Year and Date
      2010-06-18
    • Related Report
      2012 Final Research Report 2010 Annual Research Report

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Published: 2010-08-23   Modified: 2019-07-29  

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