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2013 Fiscal Year Final Research Report

Study on the integrable structure of ultradiscrete systems

Research Project

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Project/Area Number 21340034
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionThe University of Tokyo

Principal Investigator

TOKIHIRO Tetsuji  東京大学, 数理(科)学研究科(研究院), 教授 (10163966)

Co-Investigator(Renkei-kenkyūsha) OKAMOTO Kazuo  東京大学, 大学院数理科学研究科, 名誉教授 (40011720)
UIROKKUSU Rarufu  東京大学, 大学院数理科学研究科, 准教授 (20361610)
SAKAI Hidetaka  東京大学, 大学院数理科学研究科, 准教授 (50323465)
IZUMI Makoto  島根大学, 教育学部, 教授 (50263497)
TAKENAWA Tomoyuki  東京海洋大学, 海洋工学部, 准教授 (70361805)
INOUE Rei  千葉大学, 理学部, 准教授 (30431901)
Project Period (FY) 2009-04-01 – 2013-03-31
Keywordsセルオートマトン / 超離散系 / 可積分系 / 箱玉系 / 有限体 / トロピカル曲線
Research Abstract

An ultradiscrete system denotes a cellular automaton constructed from a continuous equation through a limiting procedure. We investigated the ultradiscrete systems which are considered to have integrability. The principal results in this research are; (1) the correlation functions of the box-ball systems, which are typical integrable ultrtadiscrete systems, are obtained, (2) the mathematical structure of the so-called negative solitons in the ultradiscrete KdV equation is clarified, (3) solutions to ultradiscrete Painlevé equations are constructed and a series of formulae for partition numbers are obtained, (4) integrable systems over finite fields are shown to have a property called AGR (almost good reduction).

  • Research Products

    (21 results)

All 2012 2011 2010

All Journal Article (16 results) (of which Peer Reviewed: 16 results) Presentation (4 results) Book (1 results)

  • [Journal Article] Soliton Solutions of a Generalized Discrete KdV Equation2012

    • Author(s)
      Masataka Kanki, Jun Mada and Tetsuji Tokihiro
    • Journal Title

      J. Phys. Soc. Jpn

      Volume: 81 Pages: 084002(5)

    • DOI

      10.1143/JPSJ.81.084002

    • Peer Reviewed
  • [Journal Article] A path preference cellular-automaton model for traffic flow through transit points and its application to the transcription process in biology2012

    • Author(s)
      Yoshihiro Ohta, Akinobu Nishiyama, Yoichiro Wada, Yijun Ruan, Tatsuhiko Kodama, Takashi Tsuboi, Tetsuji Tokihiro and Sigeo Ihara
    • Journal Title

      Phys. Rev. E

      Volume: 86 Pages: 021918(11)

    • DOI

      10.1103/PhysRevE.86.021918

    • Peer Reviewed
  • [Journal Article] Discrete Painleve II equation over finite fields2012

    • Author(s)
      M. Kanki, J. Mada, K. M. Tamizhmani and T. Tokihiro
    • Journal Title

      J. Phys. A : Math. Theor

      Volume: 45 Pages: 342001(8)

    • DOI

      10.1088/1751-8113/45/34/342001

    • Peer Reviewed
  • [Journal Article] Discrete Integrable Equations over Finite Fields2012

    • Author(s)
      Masataka Kanki, Jun Mada and Tetsuji Tokihiro
    • Journal Title

      SIGMA

      Volume: 8 Pages: 054(12)

    • DOI

      10.3842/SIGMA.2012.054

    • Peer Reviewed
  • [Journal Article] A classification of two-dimensional integrable mappings and rational elliptic surfaces2012

    • Author(s)
      A S Carstea and T Takenawa
    • Journal Title

      J. Phys. A : Math. Theor

      Volume: 45 Pages: 155206(15)

    • DOI

      10.1088/1751-8113/45/15/155206

    • Peer Reviewed
  • [Journal Article] Direct ultradiscretization of Ai and Bi functions and special solutions to Painelevé II equation2012

    • Author(s)
      S. Isojima, J. Satsuma and T. Tokihiro
    • Journal Title

      J. Phys. A : Math. Theor

      Volume: 45 Pages: 155203(13)

    • DOI

      10.1088/1751-8113/45/15/155203

    • Peer Reviewed
  • [Journal Article] Integrable structure of box-ball systems : crystal, Bethe ansatz, ultradiscretization and tropical geometry2012

    • Author(s)
      Rei Inoue, Atsuo Kuniba and Taichiro Takagi
    • Journal Title

      J. Phys. A : Math.Theor

      Volume: 45 Pages: 073001(64)

    • DOI

      10.1088/1751-8113/45/7/073001

    • Peer Reviewed
  • [Journal Article] Discretising the Painlev´e equations ‵a la Hirota-Mickens2012

    • Author(s)
      B. Grammaticos, A. Ramani, J. Satsuma and R. Willox
    • Journal Title

      J. Math. Phys

      Volume: 53 Pages: 023506(24)

    • DOI

      10.1063/1.3682240

    • Peer Reviewed
  • [Journal Article] Two point correlation functions for a periodic box-ball system2011

    • Author(s)
      J. Mada and T. Tokihiro
    • Journal Title

      SIGMA

      Volume: 7 Pages: 027(16)

    • DOI

      10.3842/SIGMA.2011.027

    • Peer Reviewed
  • [Journal Article] Existence and non-existence of global solutions for a discrete semilinear heat equation2011

    • Author(s)
      K. Matsuya and T. Tokihiro
    • Journal Title

      Discrete and Continuous Dynamical Systems

      Volume: 31 Pages: 209-220

    • DOI

      10.3934/dcds.2011.31.209

    • Peer Reviewed
  • [Journal Article] Construction of an Isotropic Cellular Automaton for a Reaction-Diffusion Equation by Means of a Random Walk2011

    • Author(s)
      A. Nishiyama and T. Tokihiro
    • Journal Title

      J. Phys. Soc. Jpn

      Volume: 80, No.5 Pages: 054003(6)

    • DOI

      10.1143/JPSJ.80.054003.

    • Peer Reviewed
  • [Journal Article] FOLDING TRANSFORMATIONS AND HKY MAPPINGS2011

    • Author(s)
      B. GRAMMATICOS, A. RAMANI and R. WILLOX
    • Journal Title

      Journal of Nonlinear Mathematical Physics

      Volume: 18 Pages: 75-85

    • DOI

      10.1142/S1402925111001179

    • Peer Reviewed
  • [Journal Article] Conserved quantities and generalized solutions of the ultradiscrete KdV equation2011

    • Author(s)
      Masataka Kanki, Jun Mada and Tetsuji Tokihiro
    • Journal Title

      J. Phys.A : Math. Theor

      Volume: 44 Pages: 145202(15)

    • DOI

      10.1088/1751-8113/44/14/145202

    • Peer Reviewed
  • [Journal Article] Modelling the morphology of migrating bacterial colonies2010

    • Author(s)
      A. Nishiyama, T. Tokihiro, M. Badoual and B. Grammaticos
    • Journal Title

      Physica D : Nonlinear Phenomena

      Volume: 239 Pages: 1573-1580

    • Peer Reviewed
  • [Journal Article] Correlation functions for a periodic box-ball system2010

    • Author(s)
      J. Mada and T. Tokihiro
    • Journal Title

      J. Phys. A : Math. Theor

      Volume: 43 Pages: 135205(15)

    • DOI

      10.1088/1751-8113/43/13/135205

    • Peer Reviewed
  • [Journal Article] Solving the ultradiscrete KdV equation2010

    • Author(s)
      Ralph Willox, Yoichi Nakata, Junkichi Satsuma, Alfred Ramani and Basile Grammaticos
    • Journal Title

      J. Phys. A : Math. Theor

      Volume: 43 Pages: 482003(7)

    • DOI

      10.1088/1751-8113/43/48/482003

    • Peer Reviewed
  • [Presentation] Ultradiscrete KdV equation and Box-Ball System2011

    • Author(s)
      Tetsuji Tokihiro
    • Organizer
      Tropical Geometry and Integrable Systems
    • Place of Presentation
      University of Glasgow
    • Year and Date
      20110707-08
  • [Presentation] On negative soliton solution to ultradiscrete KdV equation2011

    • Author(s)
      Tetsuji Tokihiro
    • Organizer
      The Seventh IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
    • Place of Presentation
      The University of Georgia, Athens, GA USA
    • Year and Date
      20110404-07
  • [Presentation] Correlation function for the periodic box-ball system2010

    • Author(s)
      Tetsuji Tokihiro
    • Organizer
      Satellite cinference of ICM 2010, Integrable Systems and Geometry
    • Place of Presentation
      Pondicherry University, India
    • Year and Date
      20100812-17
  • [Presentation] Correlation function of periodic box-ball system2010

    • Author(s)
      Tetsuji Tokihiro
    • Organizer
      China-Japan Joint Workshop on Integrable Systems
    • Place of Presentation
      Shaoxing, China
    • Year and Date
      20100107-10
  • [Book] 箱玉系の数理2010

    • Author(s)
      時弘哲治
    • Total Pages
      177
    • Publisher
      朝倉書店

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Published: 2015-07-16   Modified: 2019-06-06  

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