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

超离散方法研究可积元胞自动机及其解空间代数结构

基本信息

批准号：
09640273
负责人：
OHTA Yasuhiro
金额：
$ 1.92万
依托单位：
HIROSHIMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640273/
关键词：
Integrable System Cellular Automaton Ultradiscrete Painleve Equation 可積分法

项目摘要

(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 Painleve3. We constructed ultradiscrete limits deriving the elementary cellular automata (ECA) from diffusion equations and discussed the correspondence between ECA and differential equations.

（1997年）1.从可积元胞自动机出发，给出了Painleve方程的一种新形式.这些方程的自变量和因变量都是离散的。我们发现，他们捕捉的本质Painleve方程的行为，组织成一个聚结级联，并拥有特殊的解决方案。给出了元胞自动机可积的一个必要条件。讨论了元胞自动机的可积性概念.提出了二维Lotka-Volterra系统的元胞自动机等价模型。研究了参数取整数和有理数时系统的动力学行为。在整数参数的情况下，运动是完全规则的，导致严格的周期性运动。这在有理参数的情况下仍然成立，但对于有理初始条件，随着初始数据的分母增加，周期逐渐变长。在这种情况下，运动逐渐失去其规律性，导致在非理性数据的限制下的混沌行为。（1998年）从已知的离散形式出发，提出了一种系统的方法来构造Painleve方程的超离散形式.本文分析了两个非对称离散Painleve方程，即d-PII和q-PIII，证明了这两个方程都是自对偶的。这意味着相同的方程支配着沿着离散自变量的演化和在Schlesinger变换的作用下沿着沿着离散Painleve 3的参数的变换。构造了由扩散方程导出的初等元胞自动机的超离散极限，讨论了初等元胞自动机与微分方程的对应关系。