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Study of solutions and their singularities of nonlinear partial differential equations in the complex domain

Research Project

Project/Area Number 15K04966
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionSophia University

Principal Investigator

TAHARA Hidetoshi  上智大学, 理工学部, 教授 (60101028)

Project Period (FY) 2015-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords偏微分方程式 / 複素領域 / 特異点 / 形式解 / 非線型偏微分方程式 / 発散級数解 / 対数的特異点 / 解の一意性 / Gevrey指数 / Briot-Bouquet型 / ボレル総和法 / q-差分方程式 / 漸近解析 / q-差分方程式 / 漸近展開
Outline of Final Research Achievements

I studied nonlinear partial differential equations in the complex domain, and investigated in detail the behavior of singularities of solutions. The structure of singularities is completely clarified in the following two cases: the case of first order Briot-Bouquet type partial differential equations, and the case of first order nonlinear totally characteristic partial differential equations. In the case of higher order partial differential equations and q-difference partial differtential equations, many new findings have been obtained.

Academic Significance and Societal Importance of the Research Achievements

複素領域での常微分方程式の特異点の研究は, 20世紀初頭に一応の完成を見た。その後は, 数学や物理などの多くの分野で基本言語の一つとして活用されている。偏微分方程式の特異点の研究においても, 基礎的部分が十分に整理されれば, 多くの数学や物理などの分野において重要な役割を果たすと期待される。

Report

(6 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (32 results)

All 2020 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (4 results) Journal Article (11 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 10 results,  Open Access: 11 results,  Acknowledgement Compliant: 5 results) Presentation (16 results) (of which Int'l Joint Research: 8 results,  Invited: 9 results) Remarks (1 results)

  • [Int'l Joint Research] アルカラ大学(スペイン)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] アルカラ大学(スペイン)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] アルカラ大学(スペイン)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] アルカラ大学(スペイン)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Uniqueness of the solution of nonlinear singular first order partial differential equations2020

    • Author(s)
      H. Tahara
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: -

    • NAID

      130007928967

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A note on Gq-summability of formal solutions of some linear q-difference-differential equations2019

    • Author(s)
      H. Tahara and H. Yamazawa
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B75 Pages: 113-121

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Maillet type theorem for nonlinear totally characteristic partial differential equations2019

    • Author(s)
      A. Lastra and H. Tahara
    • Journal Title

      Mathematische Annalen

      Volume: - Issue: 3-4 Pages: 1603-1641

    • DOI

      10.1007/s00208-019-01864-x

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A formal solvability of a coupling equation for PDEs of Briot-Bouquet type2019

    • Author(s)
      Y. Okada, R. Schaefke and H. Tahara
    • Journal Title

      数理解析研究所講究録

      Volume: 2101 Pages: 91-97

    • Related Report
      2018 Research-status Report
    • Open Access / Int'l Joint Research
  • [Journal Article] On the unique solvability of nonlinear Fuchsian partial differential equations2018

    • Author(s)
      D. B. Bacani, J. E. C. Lope and H. Tahara
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 41 Pages: 225-239

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] q-Analogues of Laplace and Borel transforms by means of q-exponentials2017

    • Author(s)
      H. Tahara
    • Journal Title

      Ann. Inst. Fourier

      Volume: 67 Issue: 5 Pages: 1865-1903

    • DOI

      10.5802/aif.3124

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Unique solvability of coupling equations in holomorphic functions2016

    • Author(s)
      Y. Okada, R. Schaefke and H. Tahara
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B57 Pages: 69-78

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Generalized Poincare condition and convergence of formal solutions of some nonlinear totally characteristic equations2016

    • Author(s)
      H. Tahara
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 39 Pages: 863-883

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] On the lifespan of solutions to nonlinear Cauchy problems with small analytic data2016

    • Author(s)
      M. A. C. Tolentino, D. B. Bacani and H. Tahara
    • Journal Title

      Journal of Differential Equations

      Volume: 260 Issue: 1 Pages: 897-922

    • DOI

      10.1016/j.jde.2015.09.013

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] q-Analogue of summability of formal solutions of some linear q-difference-differential equations2015

    • Author(s)
      H. Tahara and H. Yamazawa
    • Journal Title

      Opuscula Mathematica

      Volume: 35 Issue: 5 Pages: 713-738

    • DOI

      10.7494/opmath.2015.35.5.713

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Analytic continuation of solutions of some nonlinear convolution partial differential equations2015

    • Author(s)
      H. Tahara
    • Journal Title

      Opuscula Mathematica

      Volume: 35 Issue: 5 Pages: 739-773

    • DOI

      10.7494/opmath.2015.35.5.739

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] 特異点をもつ一階非線型偏微分方程式の解の一意性について2020

    • Author(s)
      田原秀敏
    • Organizer
      2020年度日本数学会年会
    • Related Report
      2019 Annual Research Report
  • [Presentation] Uniqueness of the solution of nonlinear singular first order partial differential equations2019

    • Author(s)
      H. Tahara
    • Organizer
      Workshop FASFE19 : Formal and analytic solutions of functional equations
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Maillet type theorem for nonlinear totally characteristic partial differential equatioins2018

    • Author(s)
      H. Tahara
    • Organizer
      Conference CDDE2018「Complex differential and difference equations」
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Maillet type theorem for nonlinear totally characteristic partial differential equatioins2018

    • Author(s)
      田原秀敏
    • Organizer
      RIMS共同研究(公開型)「Various problems in algebraic analysis - microlocal analysis and asymptotic analysis - (代数解析学の諸相-超局所解析と漸近解析-)」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Gevrey type estimates of formal solutions of some totally characteristic partial differential equations2018

    • Author(s)
      田原秀敏
    • Organizer
      RIMS共同研究 (グループ型)「Formal and analytic solutions of functional equations on the complex domain(複素領域における形式解と解析解)」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] On analytic solutions of some nonilear singular partial differential equations2017

    • Author(s)
      H. Tahara
    • Organizer
      Workshop FASdiff17: Formal and Analytic Solutions of Diff. Equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Some examples of the Borel summability of formal solutions of linear partial differential equations2017

    • Author(s)
      田原秀敏
    • Organizer
      RIMS共同研究(公開型)「Microlocal analysis and asymptotic analysis(超局所解析と漸近解析)」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On the summability of formal solutions of some linear q-difference partial differential equations2016

    • Author(s)
      田原秀敏
    • Organizer
      RIMS研究集会 "超局所解析と特異摂動論の新展開"
    • Place of Presentation
      京都大学数理解析研究所(京都府・京都)
    • Year and Date
      2016-10-05
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] On the summability of formal solutions of some linear q-difference partial differential equations2016

    • Author(s)
      H. Tahara
    • Organizer
      Workshop FASPDE16 "Formal and Analytic Solutions of Partial Differential Equations"
    • Place of Presentation
      Lisbon (Portugal)
    • Year and Date
      2016-09-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Summability of formal solutions of q-difference partial differential equations2016

    • Author(s)
      田原秀敏
    • Organizer
      RIMS Workshop "Exponential asymptotics of difference and differential equations"
    • Place of Presentation
      京都大学数理解析研究所(京都府・京都)
    • Year and Date
      2016-06-07
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Generalized Poincare condition and convergence of formal solutions of some nonlinear totally characteristic equations2016

    • Author(s)
      田原秀敏
    • Organizer
      研究集会「Tokyo Journal of Mathematics 筱田記念号刊行に寄せて」
    • Place of Presentation
      上智大学 (東京都・千代田区)
    • Year and Date
      2016-03-20
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] q-Analogues of Laplace and Borel transforms with application to q-difference equations2016

    • Author(s)
      H. Tahara
    • Organizer
      International Conference on "Partial Differential Equations: General Theory and Variational Problems"
    • Place of Presentation
      Cebu (Philippines)
    • Year and Date
      2016-01-11
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On singular solutions of nonlinear singular partial differential equations in a sectorial domain2015

    • Author(s)
      田原秀敏
    • Organizer
      RIMS共同研究「複素の偏微分方程式における代数解析的方法」
    • Place of Presentation
      京都大学数理解析研究所 (京都府・京都)
    • Year and Date
      2015-12-09
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Maillet type theorems, convolution equations and multisummability of formal solutions2015

    • Author(s)
      H. Tahara
    • Organizer
      Conference "Differential and difference equations: analytic, arithmetic and Galoisian approaches"
    • Place of Presentation
      Lille (France)
    • Year and Date
      2015-10-22
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Analytic continuation of solutions to nonlinear convolution partial differential equations and its application2015

    • Author(s)
      H. Tahara
    • Organizer
      Conference "Analytic, Algebraic and Geometric Aspects of Differential Equations"
    • Place of Presentation
      Bedlewo (Poland)
    • Year and Date
      2015-09-18
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Lifespan of solutions to nonlinear Cauchy problems with small analytic data2015

    • Author(s)
      H. Tahara
    • Organizer
      EquaDiff 2015
    • Place of Presentation
      Lyon (France)
    • Year and Date
      2015-07-09
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Remarks] 複素領域での非線型偏微分方程式の解とその特異点の研究

    • URL

      http://pweb.cc.sophia.ac.jp/tahara/kaken.html

    • Related Report
      2019 Annual Research Report 2018 Research-status Report 2017 Research-status Report 2016 Research-status Report 2015 Research-status Report

URL: 

Published: 2015-04-16   Modified: 2021-02-19  

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