1998 Fiscal Year Final Research Report Summary
Study on Integrable Cellular Automaton and Algebraic Structure of Its Solution Space by Ultradiscretization Method
| Project/Area Number |
09640273
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
OHTA Yasuhiro Hiroshima University, Faculty of Engineering, Research Associate, 工学部, 助手 (10213745)
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| 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)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Integrable System / Cellular Automaton / Ultradiscrete / Painleve Equation |
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| 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.
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