2022 Fiscal Year Final Research Report
Studies on discrete quasi-integrable systems
| Project/Area Number |
18H01127
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | The University of Tokyo |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Keywords | 離散準可積分系 / co-primeness / 離散力学系 / 超離散系 / Hietarinta-Viallet方程式 / ファジーセルオートマトン |
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| Outline of Final Research Achievements |
We extended the discrete KdV equation so that it has the co-primeness property, which is considred as a quasi-integrable extension of the discrete KdV equation. Then, it was proved that the discrete equation is a two-dimensional extension of the Hietarinta-Viallet equation, and its reduction involves the Hietarinta-Viallet equation and its generalized eequations of higher orders. We also proved the relation between cluster algebras and co-primeness, which is an algebraic formulation of the singularity confinement. Furthermore, we succeeded to construct the quasi-integrable system of discrete equations with the same properties in arbitrary dimensions.
As an application, we proved the stability of the traffic flow model obtained by Fuzzification of the Rule 184 cell automaton (FCA184). The generalization of FCA184 incorporates the slow-to-start property and analytically determines the value of the density of the phase transition from the free-flow phase to the congested phase.
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| Free Research Field |
応用数学
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| Academic Significance and Societal Importance of the Research Achievements |
非線形可積分系は，一般には解くことのできない非線形な系の中で，厳密解を構成できる方程式系であり，方程式の持つ美しい代数構造や解が「見える」ことにより，純粋数学から工学まで広い分野にわたって応用されている．特に離散可積分系は，連続系を極限として含み，その応用範囲も広い．一方で，非線形系の中では特殊な系であり，ほとんどの系には可積分性はない．本研究成果は，特異点閉じ込め性という可積分性判定条件を代数的に再定式化することによって離散可積分系を一般化し，その枠を超えた新たな性質の良い離散系を構成したものであり，学術上も応用上もその意義は大きく，他分野へも影響を与えうるものと考えられる．
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