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Study of a movable singular point of a Hamiltonian system and Borel summability

Research Project

Project/Area Number 20K03683
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionHiroshima University

Principal Investigator

Yoshino Masafumi  広島大学, 先進理工系科学研究科(理), 名誉教授 (00145658)

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2022: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2021: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywordsハミルトン系 / 動く特異点 / 超級数 / ボレル総和法 / 超可積分性 / 接続問題 / 進化項をもつロトカボルテラ方程式 / バーコフ変換 / transseries の第一積分 / 非可積分ハミルトン系 / 小進化項をもつロトカボルテラ方程式 / transseriesのボレル総和法 / transseriesの第一積分 / 非可積分性 / 小進化項を持つ3種ロトカボルテラ方程式 / 大域的ボレル総和法 / 解の爆発 / 動く分岐点 / バーコフ変換理論 / 爆発解 / 非線形放物型方程式 / パンルベ性
Outline of Research at the Start

本研究では、パンルベ性を持たないハミルトン系の動く分岐点を持つ解の構成とその特異点の構造を主に研究する。この方程式達は、数理物理であらわれる非線形波動、非線形放物型、非線形シュレディンガー方程式等の爆発解の構成における自己相似解の満たす方程式として現れる。研究では、まず自由度が2の場合の動く分岐点を研究し、特異点が複数あるような解を構成・解析したのち、一般の自由度の場合を研究する。従来の研究と比較して特徴的な点は、力学系の視点からのアプローチすなわちバーコフ変換の一般化を用いて解析することにある。解析的な手法として偏微分方程式に対するボレル総和法理論を新しく構成して用いる。

Outline of Final Research Achievements

We study movable singularity for some Hamiltonian system satisfied by the radially symmetric self-similar solution of the nonlinear wave equation, semi linear heat equation and the nonlinear Schrodinger equation without the so-called Painleve property. We show that there is a movable branch point expressed by the elliptic function and a Birkhoff-type transformation. In proving the fact, we extend a Borel summability theory for a partial differential equation. As an application of the asymptotic theory, we find the new behavior of a solution of the three-species Lotka-Volterra model with an evolutional effect.

Academic Significance and Societal Importance of the Research Achievements

研究成果の社会的意義は、研究の対象となる方程式達が数理物理での基礎方程式であり、量子論、レーザーなど社会の多くの分野で応用されており、それらに新しい知見を与えた点にある。学術的意義は、今回の研究成果を従来の研究と比較したとき、動く分岐点の存在がバーコフ型変換を用いて楕円関数からの変換によって引き起こされることが示されたこと、さらに証明も解析分野の結果であるボレル総和法を基礎にした見通しの良い議論になっているという点にある。証明で示された偏微分方程式に対する発散解の構成とボレル総和法理論の拡張もボレル総和法分野での新しい応用例を与えた。

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (16 results)

All 2024 2023 2022 2021 2020 Other

All Journal Article (8 results) (of which Peer Reviewed: 8 results,  Open Access: 1 results) Presentation (7 results) (of which Int'l Joint Research: 5 results,  Invited: 5 results) Remarks (1 results)

  • [Journal Article] Summability of Transseries Solution of Non-integrable Hamiltonian System2024

    • Author(s)
      Masafumi Yoshino
    • Journal Title

      Journal of Dynamical and Control Systems

      Volume: - Issue: 2

    • DOI

      10.1007/s10883-024-09692-2

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Eco-evolutionary feedback as a driver of periodic state shifts in tri-trophic food chains2023

    • Author(s)
      Yoshinari Tanaka and Masafumi Yoshino
    • Journal Title

      Evolutionary Ecology

      Volume: - Issue: 6 Pages: 1001-1020

    • DOI

      10.1007/s10682-023-10278-w

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Solution with Movable Singular Points of Some Hamiltonian System2023

    • Author(s)
      Masafumi Yoshino
    • Journal Title

      Contemporary Mathematics

      Volume: 782 Pages: 207-218

    • DOI

      10.1090/conm/782/15730

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Movable Singularity of Some Hamiltonian System Related with Blowup Phenomenon2023

    • Author(s)
      Masafumi Yoshino
    • Journal Title

      Publications of RIMS

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] First Integral of Non-integrable Hamiltonian System2023

    • Author(s)
      Masafumi Yoshino
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Solution with Movable Singular Points of Some Hamiltonian System2022

    • Author(s)
      Masafumi Yoshino
    • Journal Title

      Proceedings of "Formal and Analytic Solutions of Differential Equations", Contempoary Mathematics

      Volume: -

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Global Borel summability of some partial differential equation2021

    • Author(s)
      Masafumi Yoshino
    • Journal Title

      Proceedings of "Formal and Analytic Solutions of Differential Equations", World Scientific Publishing Europe Ltd.

      Volume: -

    • DOI

      10.1142/q0335

    • ISBN
      9781800611351, 9781800611368
    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Global Borel summability of some partial differential equation2021

    • Author(s)
      Masafumi Yoshino
    • Journal Title

      Proceedings of "Formal and Analytic Solutions of Differential Equations, World Scientific Publishing Europe Ltd.

      Volume: -

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] Summability and global property of transseries solution of Hamiltonian systems2023

    • Author(s)
      Masafumi Yoshino
    • Organizer
      COMPLEX DIFFERENTIAL AND DIFFERENCE EQUATIONS II (Banach center)
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Connection problem for transseries solution and first integral2023

    • Author(s)
      Masafumi Yoshino
    • Organizer
      Polish-Japanese workshop on differential equations in the complex domain (Banach center)
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Movable singular point of solution of some Hamiltonian system2022

    • Author(s)
      Masafumi Yoshino
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2022 Research-status Report
  • [Presentation] Movable singular point of non autonomous Hamiltonian system of degree of freedom one2022

    • Author(s)
      Masafumi Yoshino
    • Organizer
      日本数学会年会
    • Related Report
      2021 Research-status Report
  • [Presentation] Multiple movable singularity of some Hamiltonian system -Application of Borel summability-2021

    • Author(s)
      Masafumi Yoshino
    • Organizer
      FASnet21 -Alcala Spain-
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Singular solution of non-integrable Hamiltonian system2021

    • Author(s)
      Masafumi Yoshino
    • Organizer
      Exact WKB Analysis, Microlocal Analysis, Painleve Equations and Related Topics
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Parametric Borel summability of a certain PDE2020

    • Author(s)
      Masafumi Yoshino
    • Organizer
      FASdiff20 (Alcala, Spain)
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks]

    • URL

      https://home.hiroshima-u.ac.jp/yoshinom/

    • Related Report
      2020 Research-status Report

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Published: 2020-04-28   Modified: 2025-01-30  

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