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On the exact WKB method from a viewpoint of microlocal analysis

Research Project

Project/Area Number 23540178
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionHokkaido University

Principal Investigator

HONDA Naofumi  北海道大学, 理学(系)研究科(研究院), 准教授 (00238817)

Co-Investigator(Kenkyū-buntansha) UCHIDA Motoo  大阪大学, 理学研究科, 准教授 (10221805)
Project Period (FY) 2011 – 2013
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2011: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Keywords完全WKB解析 / ストークス現象 / 代数解析 / 超局所解析 / 完全WKB法 / インスタントン解 / パンルベ階層
Research Abstract

We study Stokes pheonmenon for a higher order linear differential equation with a large parameter, and we also study the same problems for a non-linear differential equation such as a Painlev'e hierarchy.
A Stokes geometry for a higher order linear differential equation is quite different from one for a 2nd order linear differential equation becuase of existence of virtual turning points and new Sotkes curves. It is very complicated, and thus, possibility to succesively obtain a Stokes coefficient on each Stokes curve is quite uncertain. By using so called the depth function, in this study, we have shown that it is always possible to have all the Stokes coefficients succesively.
We also have succeeded in constructing an instanton-type solution for the first Painlev'e hierarchy (PI)_m. This result is quite important becuase it contains sufficiently many free parametners, and hence, we can take a family of these solutions as a basis of solutions for a connection problem.

Report

(4 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Research-status Report
  • 2011 Research-status Report
  • Research Products

    (24 results)

All 2014 2013 2012 2011 Other

All Journal Article (18 results) (of which Peer Reviewed: 14 results) Presentation (6 results) (of which Invited: 2 results)

  • [Journal Article] Joyce McLaughlin and Gen Nakamura, Conditional stability for single interior measurement2014

    • Author(s)
      Naofumi Honda
    • Journal Title

      Inverse Problems

      Volume: 30(055001) Pages: 1-19

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] A computer-assisted study of the Landau- Nakanishi Geometry2014

    • Author(s)
      Naofumi Honda and Takahiro Kawai
    • Journal Title

      RIMS Kokyuroku

      Volume: No.1861 Pages: 100-110

    • Related Report
      2013 Final Research Report
  • [Journal Article] Conditional stability for single interior measurement2014

    • Author(s)
      Naofumi Honda, Joyce McLaughlin and Gen Nakamura
    • Journal Title

      Inverse Problems

      Volume: 30 Pages: 1-19

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A computer-assisted study of the Landau-Nakanishi Geometry2014

    • Author(s)
      Naofumi Honda, Takahiro Kawai
    • Journal Title

      RIMS Kokyuroku

      Volume: 1861 Pages: 100-110

    • Related Report
      2013 Annual Research Report
  • [Journal Article] On kernel functions and symbols of analytic pseudo-differential operator2013

    • Author(s)
      Takashi Aoki, Naofumi Honda and Susumu Yamazaki
    • Journal Title

      RIMS Koukyuroku

      Volume: No.1835 Pages: 21-37

    • Related Report
      2013 Final Research Report
  • [Journal Article] On the number of the turning points of the second kind of the Noumi- Yamada systems with a large parameter2013

    • Author(s)
      Takashi Aoki, Naofumi Honda and Yoko Umeta
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B37 Pages: 1-30

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] On a construction of general formal solutions for equations of the first Painleve hierarchy I2013

    • Author(s)
      Takashi Aoki, Naofumi Honda, Yoko Umeta
    • Journal Title

      Advances in Mathematics

      Volume: 235 Pages: 496-524

    • Related Report
      2013 Annual Research Report 2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Multi-specialization and multi-asymptotic expansions2013

    • Author(s)
      Naofumi Honda and Luca Prelli
    • Journal Title

      Advances in Mathematics

      Volume: 232 Pages: 432-498

    • NAID

      120006459687

    • Related Report
      2013 Annual Research Report 2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators2013

    • Author(s)
      Naofumi Honda, Gen Nakamura and Murad Sini
    • Journal Title

      Mathematische Annalen

      Volume: 355 Pages: 401-427

    • Related Report
      2013 Annual Research Report 2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] On kernel functions and symbols of analytic pseudo-differential operators2013

    • Author(s)
      T.Aoki, N.Honda and S.Yamazaki
    • Journal Title

      RIMS Koukyuroku

      Volume: 1835 Pages: 21-37

    • Related Report
      2013 Annual Research Report
  • [Journal Article] On the number of the turning points of the second kind of the Noumi-Yamada systems with a large parameter2013

    • Author(s)
      T. Aoki, N. Honda and Y. Umeta
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B37 Pages: 1-30

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the sheaf of Laplace hyperfunctions with holomorphic parameters2012

    • Author(s)
      Naofumi Honda and Kohei Umeta
    • Journal Title

      J. Math. Sci. Univ. Tokyo

      Volume: 19 Pages: 559-586

    • NAID

      120005678225

    • Related Report
      2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Geometric properties of the Riemann surfaces associated with the Noumi- Yamada systems with a large parameter2011

    • Author(s)
      Takashi Aoki and Naofumi Honda
    • Journal Title

      J. Math. Soc. Japan

      Volume: 63(4) Pages: 1085-1119

    • NAID

      10030344748

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Stratified Whitney jets and termpered ultradistributions on the subanalytic site2011

    • Author(s)
      Naofumi Honda and Giovanni Morando
    • Journal Title

      Bulletin de la Société Mathématique de France

      Volume: 139(3) Pages: 389-435

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Multi-specialization and multi-asymptotic expansions2011

    • Author(s)
      Naofumi Honda and Luca Prelli
    • Journal Title

      Proc. Japan. Acad

      Volume: 87(A) Pages: 69-72

    • NAID

      120006459687

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Stratified Whitney jets and termpered ultradistributions on the subanalytic site2011

    • Author(s)
      N. Honda and G. Morando
    • Journal Title

      Bulletin de la Soci\'et\'e Math\'ematique de France

      Volume: 130 Pages: 389-435

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Geometric properties of the Riemann surfaces associated with the Noumi-Yamada systems with a large parameter2011

    • Author(s)
      T. Aoki and N. Honda
    • Journal Title

      J. Math. Soc. Japan

      Volume: 63 Pages: 1085-1119

    • NAID

      10030344748

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Multi-specialization and multi-asymptotic expansions2011

    • Author(s)
      N. Honda and L. Prelli
    • Journal Title

      Proc. Japan Acad

      Volume: 87 Pages: 69-72

    • NAID

      120006459687

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Presentation] 多重超局所化について, 代数解析学と局所凸空間2014

    • Author(s)
      本多 尚文
    • Place of Presentation
      日本大学(東京都千代田区)
    • Year and Date
      2014-02-18
    • Related Report
      2013 Final Research Report
  • [Presentation] Conditional stability for single interior measurement, 代数解析学の諸問題2013

    • Author(s)
      本多 尚文
    • Place of Presentation
      千葉大学(千葉県千葉市)
    • Year and Date
      2013-11-30
    • Related Report
      2013 Final Research Report
  • [Presentation] Multi-specialization and multi-microlocalization, Recent trends in Algebraic Analysis2013

    • Author(s)
      Naofumi Honda
    • Place of Presentation
      Univ. Padova (Padova, Intaly)
    • Year and Date
      2013-02-19
    • Related Report
      2013 Final Research Report
  • [Presentation] Micro-analytic Smatrix theory revisited, Recent development of microlocal analysis and asympotitic analysis2012

    • Author(s)
      Naofumi Honda
    • Place of Presentation
      京都大学数理解析研究所(京都市)
    • Year and Date
      2012-10-24
    • Related Report
      2013 Final Research Report
  • [Presentation] 多重超局所化について

    • Author(s)
      本多 尚文
    • Organizer
      代数解析学と局所凸空間
    • Place of Presentation
      日本大学(東京都千代田区)
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Conditional stablity for single interior mesurement

    • Author(s)
      本多 尚文
    • Organizer
      代数解析学の諸問題
    • Place of Presentation
      千葉大学(千葉県千葉市)
    • Related Report
      2013 Annual Research Report
    • Invited

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Published: 2011-08-05   Modified: 2019-07-29  

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