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

(1997) 1。从可积元胞自动机出发，提出了一种新的Painleve方程形式。这些方程在自变量和因变量上都是离散的。我们证明了它们捕捉了Painleve方程行为的本质，将自己组织成一个合并级联，并具有特殊的解。给出了元胞自动机可积性的一个必要条件。讨论了元胞自动机的可积性概念。我们提出了二维Lotka-Volterra系统的元胞自动机。研究了参数取值为整数和有理值时的动力学特性。在整数参数的情况下，运动是完全规则的，导致严格的周期运动。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，导致在非理性数据极限下的混沌行为。我们提出了一种系统的方法，从已知的离散形式出发，构造Painleve方程的超离散版本。我们分析了两个非对称离散疼痛级方程，即d-PII和q-PIII。我们证明了两个方程都是自对偶的。这意味着同一方程支配着沿离散自变量的演化和沿离散Painleve3参数施莱辛格变换作用下的变换。从扩散方程出发，构造了初等元胞自动机的超离散极限，并讨论了初等元胞自动机与微分方程的对应关系。

(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.