Study on cellular automata derived from discrete integrable systems

• Project/Area Number: 26400109
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: The University of Tokyo
• Principal Investigator: Tokihiro Tetsuji 東京大学, 大学院数理科学研究科, 教授 (10163966)
• Research Collaborator: KANKI Masataka ; MASE Takafumi ; KAMIYA Ryo
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 準可積分系 / co-primeness条件 / 特異点閉じ込め / Laurent性 / 代数的エントロピー / 超離散 / 超離散系 / coprimeness / LP代数 / 離散可積分系 / 離散戸田方程式 / 線形化可能系 / co-primeness / 戸田格子方程式 / 特異値閉じ込め / 離散KdV方程式 / Hietarinta-Viallet方程式 / 得意点閉じ込め

Outline of Final Research Achievements

We reformulated the singularity confinement property as the co-primeness property of the discrete mappings. Typical discrete integrable systems including higher dimensional discrete soliton equations are proved to satisfy this co-primeness property. We proposed the notion of the discrete quasi-integrable equation which has the co-primeness property but is not integrable such as the Hietarinta-Viallet equation. Higher dimensional analogue of the Hietarinta-Viallet equation and quasi-integrable analogue of the discrete Toda lattice equation and its hegher dimensional extensions were constructed. We also investigated their mathematical structure by showing the Laurent property of the lower dimensional equations reduced from the quasi-integrable quations and by rigorously estimating their algebraic entropy.

Academic Significance and Societal Importance of the Research Achievements

特異点閉じ込めは，坂井によって，特異点解消による双有理写像の構成として幾何学的に説明された．本研究は，この特異点閉じ込めを代数的に再定式化したものであり，学術的な意義は大きい．実際に，この定式化によって，新たに離散準可積分系と呼ばれるよい数理構造をもつ力学系のクラスを定式化でき，多くのよい数学的性質を持ちながら可積分ではない系を特徴づけることができた．今後は，この離散準可積分系の分類問題に取り組まなければならないが，そのための数学的な準備や手法の開発は，当該数理科学分野において大きな意味を持つと思われる．

Research Products

• [Int'l Joint Research] NIPNE-HH(Romania)
• [Journal Article] On the Coprimeness Property of Discrete Systems without the Irreducibility Condition (2018)
  • Author(s): Kanki Masataka、Mase Takafumi、Tokihiro Tetsuji
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: 14 Pages: 065-065
  • DOI: 10.3842/sigma.2018.065
  • Peer Reviewed / Open Access
• [Journal Article] A two-dimensional lattice equation as an extension of the Heideman-Hogan recurrence (2018)
  • Author(s): Ryo Kamiya, Masataka Kanki, Takafumi Mase, and Tetsuji Tokihiro
  • Journal Title: J. Phys. A: Math. Theor. Volume: 51 Issue: 12 Pages: 125203-125203
  • DOI: 10.1088/1751-8121/aaad47
  • Peer Reviewed
• [Journal Article] Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation (2017)
  • Author(s): Ryo Kamiya, Masataka Kanki, Takafumi Mase, Tetsuji Tokihiro
  • Journal Title: Journal of Mathematical Physics Volume: 58 Issue: 1 Pages: 012702-012702
  • DOI: 10.1063/1.4973744
  • Peer Reviewed
• [Journal Article] Graphs emerging from the solutions to the periodic discrete Toda equation over finite fields (2016)
  • Author(s): Masataka Kanki, Yuki Takahashi and Tetsuji Tokihiro
  • Journal Title: Nonlinear Theory and Its Applications, IEICE Volume: 7 Issue: 3 Pages: 338-353
  • DOI: 10.1587/nolta.7.338
  • NAID: 130005161833
  • ISSN: 2185-4106
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Singularity confinement and chaos in two-dimensional discrete systems (2016)
  • Author(s): Masataka Kanki, Takafumi Mase, Tetsuji Tokihiro
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 49 Issue: 23 Pages: 23LT01-23LT01
  • DOI: 10.1088/1751-8113/49/23/23lt01
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Algebraic entropy of an extended Hietarinta-Viallet equation (2015)
  • Author(s): Masataka Kanki, Takafumi Mase and Tetsuji Tokihiro
  • Journal Title: J. Phys. A: Math. Theor. Volume: 48 Issue: 35 Pages: 355202-355202
  • DOI: 10.1088/1751-8113/48/35/355202
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Integrability criterion in terms of cop rime property for the discrete Toda equation (2015)
  • Author(s): M. Kanki, J. Mada, T. Tokihiro
  • Journal Title: Journal of Mathematical Physics Volume: 56 Issue: 2 Pages: 022706-022706
  • DOI: 10.1063/1.4908109
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Coupled discrete KdV equations and modular genetic networks (2015)
  • Author(s): A. S. Carstea and T. Tokihiro
  • Journal Title: J. Phys. A: Math. Theor. Volume: 48 Issue: 5 Pages: 055205-055205
  • DOI: 10.1088/1751-8113/48/5/055205
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Irreducibility and co-primeness as an integrability criterion for discrete equations (2014)
  • Author(s): Masataka Kanki, Jun Mada, Takafumi Mase and Tetsuji Tokihiro
  • Journal Title: J. Phys. A: Math. Theor. Volume: 47
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Coprimeness condition as an algebraic reinterpretation of singularity confinement (2016)
  • Author(s): 時弘哲治
  • Organizer: 応用数理学会
  • Place of Presentation: 北九州国際会議場
  • Year and Date: 2016-09-12
  • Invited
• [Presentation] On quasi-integrable extension of 2D Toda equation (2016)
  • Author(s): Tetsuji Tokihiro
  • Organizer: The 3rd China-Japan Joint Workshop on Integrable Systems
  • Place of Presentation: Shaanxi Normal University, Xi’an (China)
  • Year and Date: 2016-08-19
  • Int'l Joint Research / Invited
• [Presentation] Coprimeness as a quasi-integrability criterion for discrete equations (2015)
  • Author(s): Tetsuji Tokihiro
  • Organizer: Workshop "Topics on tropical geometry, integrable systems and positivity"
  • Place of Presentation: 青山学院大学
  • Year and Date: 2015-12-22
  • Invited
• [Presentation] Irreducibility and co-primeness of terms in discrete equations with respect to initial variables, (2014)
  • Author(s): Tetsuji Tokihiro
  • Organizer: Three days on Painleve equations and their applications
  • Place of Presentation: Roma (Italy)
  • Year and Date: 2014-12-18 – 2014-12-20
  • Invited
• [Presentation] Integrable systems over finite fields (2014)
  • Author(s): Tetsuji Tokihiro
  • Organizer: International Conference on Symmetries and Integrability in Difference Equations
  • Place of Presentation: Indian Institute of Science (India)
  • Year and Date: 2014-07-16 – 2014-07-21
  • Invited
• [Presentation] Ultradiscrete Systems (2014)
  • Author(s): Tetsuji Tokihiro
  • Organizer: Workshop on Discrete Integrable Systems
  • Place of Presentation: Indian Institute of Science (India)
  • Year and Date: 2014-07-09 – 2014-07-14
  • Invited