Discrete and Ultradiscrete integrable systems in terms of the theory of number theoretic dynamical systems

• Project/Area Number: 17K14211
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kansai University
• Principal Investigator: KANKI Masataka 関西大学, システム理工学部, 准教授 (20755897)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); 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)
• Keywords: 可積分系 / 漸化式 / 超離散系 / 力学系 / 離散力学系 / 離散可積分系 / 代数的エントロピー / 離散KdV方程式 / 格子力学系 / 可積分性判定 / 特異点閉じ込め / 戸田格子 / ローラン現象 / coprimeness / 数理物理 / 関数方程式論

Outline of Final Research Achievements

The aim of this research is to rigorously re-define the integrability of discrete dynamical systems through the study of the algebraic and the number theoretic aspects of difference equations.
An elaboration on the integrability criteria helps us to define the "integrability" of multi-dimensional lattice systems and the systems defined over a number theoretic field. We have defined several new types of difference equations, which are considered to be "partially" integrable in terms of our integrability criterion called the "co-primeness" condition.
We expect that this study leads us to a novel perspective in the field of integrable systems and mathematical physics.

Academic Significance and Societal Importance of the Research Achievements

可積分系の研究には長い歴史があるが、離散系における可積分性についての厳密な取り扱いは発展途上である。本研究はこのテーマについて、従来の方程式を拡張した系に適応できる新しい可積分性判定基準として「互いに素条件」を導入した。
またこれらの基準の意味するところを、既知の判定基準と比較検討することで一見単純に思えるが難しい漸化式の世界の複雑さを解き明かすための準備となる研究を行うことができた。

Research Products

• [Journal Article] 可積分性判定 (2019)
  • Author(s): 神吉雅崇
  • Journal Title: 数理科学 Volume: 674
• [Journal Article] 多次元格子上の擬似可積分系 (2018)
  • Author(s): 神吉雅崇, 時弘哲治, 間瀬崇史
  • Journal Title: 数理解析研究所講究録 Volume: 2071 Pages: 17-39
• [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] Toda type equations over multi-dimensional lattices (2018)
  • Author(s): Kamiya Ryo、Kanki Masataka、Mase Takafumi、Okubo Naoto、Tokihiro Tetsuji
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 51 Issue: 36 Pages: 364002-364002
  • DOI: 10.1088/1751-8121/aad375
  • Peer Reviewed
• [Journal Article] Nonlinear forms of coprimeness preserving extensions to the Somos-4 recurrence and the two-dimensional Toda lattice equation?investigation into their extended Laurent properties (2018)
  • Author(s): Kamiya Ryo、Kanki Masataka、Mase Takafumi、Tokihiro Tetsuji
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 51 Issue: 35 Pages: 355202-355202
  • DOI: 10.1088/1751-8121/aad074
  • Peer Reviewed
• [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
• [Presentation] ある多項間漸化式の代数的エントロピーについて (2020)
  • Author(s): 神吉雅崇
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] Coprimenesspreserving extensions to discrete integrable systems (2019)
  • Author(s): Masataka Kanki
  • Organizer: 9th International Congress on Industrial and Applied Mathematics
  • Int'l Joint Research
• [Presentation] Coprimeness property of the Toda type equations over multi-dimensional lattices (2018)
  • Author(s): Masataka Kanki
  • Organizer: Symmetry and Integrability of Difference Equations 13 in Fukuoka
  • Int'l Joint Research
• [Presentation] 離散可積分性判定と互いに素条件 (2018)
  • Author(s): 神吉雅崇
  • Organizer: 可積分系理論から見える数理構造とその応用
  • Invited
• [Presentation] 漸化式と互いに素条件 (2018)
  • Author(s): 神吉雅崇
  • Organizer: 2018年度函数方程式論サマーセミナー
• [Presentation] 疑似可積分性をもつ離散方程式 (2017)
  • Author(s): 神谷 亮、神吉 雅崇、時弘 哲治、間瀬 崇史
  • Organizer: 日本応用数理学会年会
• [Presentation] 離散力学系の可積分性判定について-線形化可能系を中心に- (2017)
  • Author(s): 神吉 雅崇
  • Organizer: 数理科学の拡がり：可積分系・数理医学
• [Presentation] 離散可積分系の判定手法について (2017)
  • Author(s): 神吉 雅崇
  • Organizer: 函数方程式論サマーセミナー