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Study on Integrable Cellular Automaton and Algebraic Structure of Its Solution Space by Ultradiscretization Method

Research Project

Project/Area Number 09640273
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionHIROSHIMA UNIVERSITY

Principal Investigator

OHTA Yasuhiro  Hiroshima University, Faculty of Engineering, Research Associate, 工学部, 助手 (10213745)

Co-Investigator(Kenkyū-buntansha) KATO Hiroko  Hiroshima University, Faculty of Engineering, Research Associate, 工学部, 助手 (60284171)
ITO Masaaki  Hiroshima University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (10116535)
SHIBA Masakazu  Hiroshima University, faculty of Engineering, Professor, 工学部, 教授 (70025469)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsIntegrable System / Cellular Automaton / Ultradiscrete / Painleve Equation / 可積分法
Research Abstract

(1997)
1. Starting from integrable cellular automata we presented a novel form of Painleve equations. These equations are discrete in both the independent variable and the dependent one. We showed that they capture the essence of the behavior of the Painleve equations, organize themselves into a coalescence cascade and possess special solutions. A necessary condition for the integrability of cellular automata was also presented. We discussed about the notion of integrability of the cellular automata.
2. We presented a cellular automaton equivalent for the two-dimensional Lotka-Volterra system. The dynamics was studied for integer and rational values of the parameters. In the case of integer parameters the motion is perfectly regular leading to strictly periodic motion. This is still true in the case of rational parameters, but for rational initial conditions the period becomes progressively longer as the denominator of the initial data increases. The motion, in this case, progressively loses its regularity resulting in chaotic behavior in the limit of irrational data.
(1998)
1. We presented a systematic way to construct ultra-discrete versions of the Painleve equations starting from known discrete forms.
2. We analysed two asymmetric discrete Painleve equations, namely d-PII and q-PIII.We showed that both equations are self-dual. This means that the same equation governs the evolution along the discrete independent variable and the transformations under the action of the Schlesinger transforms along the parameters of the discrete Painleve
3. We constructed ultradiscrete limits deriving the elementary cellular automata (ECA) from diffusion equations and discussed the correspondence between ECA and differential equations.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (48 results)

All Other

All Publications (48 results)

  • [Publications] B.Grammaticos: "A Miura of the Painleve I Equation and Its Discrete Analogs" Lett.Math.Phys.39. 179-186 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] B.Grammaticos: "Cellular Automata and Ultra-Discrete Painleve Equations" Phys.Lett.A. 226. 53-58 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Kajiwara: "Rational solutions for the Discrete Painleve II Equation" Phys.Lett.A. 232. 189-199 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] R.Hirota: "From Integrability to Chaos in a Lotka-Volterra Cellular Automaton" Phys.Lett.A. 236. 39-44 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] D.Takahashi: "Constructing Solutions to the Ultradiscrete Painleve Equations" J.Phys.A. 30. 7953-7966 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] G.Schmieder: "Realisierungen des Idealen Randes einer Riemannschen Fluche unter Konformen Abschliβungen" Arch.Math.68. 36-44 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] G.Schmieder: "Realizations of the ideal boundary of a planar Riemann surface m the sphere" Nonlinear Analysis,Theory,Methods & Applications. 30. 5267-5274 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 加藤比呂子: "多変量時系列モデルによる経済システムの動的解析" 統計数理. 45. 301-318 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Sasa: "Bilinear Form Approach to the Self-Dual Yang-Mills Equations and Integrable Systems in (2_<T1>) Dimension" J.Phys.Soc.Jpn.67. 83-86 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Ramani: "Self-Duality and Schlesinger chains for the Asymmetric dP_<II> and q-P_<III> Equations" Comm.Math.Phys.192. 67-76 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Kajiwara: "Determinant Structure of the Rational Solutions for the Painleve IV Equation" J.Phys.A. 31. 2431-2446 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] B.Grammaticos: "Degeneration through Coalescence of the q-Painleve VI Equation" J.Phys.A. 31. 3545-3558 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Ramani: "The Ultimate Discretisation of the Painleve Equations" Physica D. 114. 185-196 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] G.Schmieder: "One-parametar variations of the ideal boundary and compact continuations of a Riemann surface" Analysis. 18. 125-130 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Ito: "Area theorems for conformal mapping and Rankine ovoids" Computational Methods and Function Theory(CMFT'97). 1-9 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kato: "An Application of the Bagesian Time Series Model and Statistical System Analysis for FO Control" Speech Comm.24. 325-339 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Ohta: "Determinant and Pfaffian Solutions for Discrete Soliton Equations" CRM Proceedings and Lecture Notes Series. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.J.Ablowitz: "On Discretizations of the Vector Nonlinear Schrodinger Equation" Phys.Lett.A. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] B.Grammaticos: "A Miura of the Painleve I Equation and Its Discrete Analogs" Lett.Math.Phys.39. 179-186 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] B.Grammaticos: "Cellular Automata and Ultra-Discrete Painleve Equations" Phys.Lett.A 226. 53-58 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Kajiwara: "Rational Solutions for the Discrete Painleve II Equation" Phys.Lett.A 232. 189-199 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] R.Hirota: "From Integrability to Chaos in a Lotka-Volterra Cellular Automaton" Phys.Lett.A 236. 39-44 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] D.Takahashi: "Constructing Solutions to the Ultradiscrete Painleve Equations" J.Phys.A 30. 7953-7966 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] G.Schmieder: "Realisierungen des Idealen Randes einer Riemannschen Flache unter konformen Abschliebetaungen" Archiv der Mathematik. 68. 36-44 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] G.Schmieder: "Realizations of the ideal boundary of a planar Riemann sur-face in the sphere" Nonlinear Analysis, Theory, Meth-ods & Applications. 30. 5267-5274 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kato: "Dynamical Analysis for Economic System by Multivariate Time Series Model" Proceedings of the Institute of Sta-tistical Mathematics. 45. 301-318 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Sasa: "Bilinear Form Approach to the Self-Dual Yang-Mills Equa-tions and Integrable Systems in (2+1) -Dimension" J.Phys.Soc.Jpn.67. 83-86 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Ramani: "Self-Duality and Schlesinger Chains for the Asymmetric d-P_<II> and q-P_<III> Equations" Comm.Math.Phys.192. 67-76 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Kajiwara: "Determinant Structure of the Rational Solutions for the Painleve IV Equation" J.Phys.A 31. 2431-2446 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] B.Grammaticos: "Degeneration through Coalescence of the q-Painleve VI Equation" J.Phys.A 31. 3545-3558 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Ramani: "The Ultimate Discretisation of the Painleve Equations" Physica D. 114. 185-196 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] G.Schmieder: "One-parameter variations of the ideal boundary and com-pact continuations of a Riemann surface" Analysis. 18. 125-130 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.Ito: "Area theorems for conformal mapping and Rankine ovoids" Computational Methods and Func-tion Theory (CMFT'97). 1-9 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Kato: "An Application of the Baysian Time Series Model and Sta-tistical System Analysis for F0 Control" Speech Comm.24. 325-339 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Ohta: "Determinant and Pfaffian solutions for Discrete Soliton Equations" CRM Proceedings and Lecture Notes Series. (to appear.).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] M.J.Ablowitz: "On Discretizations of the Vector Nonlinear Schrodinger Equation" Phys.Lett.A. (to appear.).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Ramani: "The ultimate discretination of the Painleve equations" Physica D. 114. 185-196 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] B.Grammatics: "Degeneration through Coalescence of the q-Painleve VI Equation" J.Phys.A. 31. 3545-3558 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Kajiwara: "Determinant Structure of the Rational Solutions for the Painleve IV Equation" J.Phys.A. 31. 2431-2446 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Ramani: "Self-Duality and Schlesinger Chains for the Asymmetric d-P_<II> and q-P_<II> Equations" Comm.Math.Phys.192. 67-76 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Ito: "Area theorems for conformal mapping and Rankine ovoids" Computational Methods and Function Theory. to appear.

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Kato: "An Application of Bayesian Time Series Model and Statistical System Analysis for FO Control" J.Speech Comm.24. 325-339 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] B.Grammaticos: "Cellular automato and ultra-discrete Painlerie eequations" Phys,Lett,A. 226. 53-58 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] R.Hirota: "From mtegrability to chaos in a Lotka-Volterva cellular automaton" Phys,Lett,A. 236. 39-44 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] B.Grammaticos: "A Miura of the Painleve I Equation and Its Dissrete Analogs" Lett.Math,Phys,. 39. 179-186 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] D.Schmieder: "Realisierungen des Idealen Randes einer Riemannschen Flache unter konformen Arschlie Bungen" Archiv der Mathematik. 68. 36-44 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] G.Schmieder: "One-parameter variations of the ideal boundarg and compact continuations of a Ricmarn snrface" Aralysio. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] 加藤 比呂子: "多変量時系列モデルによる経済システムの動的解析" 統計数理. 45. 303-321 (1998)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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