Euler-Lagrange correspondence of ultra-discrete systems

众所周知，Burgers元胞自动机包括Rule184初等元胞自动机，它是基本的交通流模型。Burgers CA是一个欧拉表示方程，其中变量表示一个场的强度。最近发现了规则184的拉格朗日表示，其中变量表示粒子的位置。由于这种表示是以Max-Plus代数的形式表示的，因此考虑两种表示之间的联系是很自然的。在“交通流模型的元胞自动机的欧拉-拉格朗日对应”的论文中，matsukidaira和KNishinari，物理学报，vol .90,No. 1。8,p088701（2003）”，我们已经找到了欧拉表示和拉格朗日表示之间的变换公式，我们可以利用Max-Plus代数和阶跃函数之间的一个新的代数公式来得到这个公式。我们还将该方法应用于多值、多速度的交通流模型，得到了Fukui-Ishibashi模型和快速启动模型的欧拉-拉格朗日对应关系。此外，我们还将该方法应用于孤立元胞自动机，成功地找到了Box和Ball系统的欧拉-拉格朗日对应关系。我们现在正在为这个结果准备论文。

It is well known that Burgers cellular automaton includes Rule184 elementary cellular automaton, which is the basic traffic flow model Burgers CA is an Euler representation equation, in which variable expresses a strength of a field. Recently, Lagrange representation of Rule 184 has been found, in which variable expresses a position of particles. As this representation is expressed in a form of Max-Plus algebra, it is natural to consider the link between two representations. In the paper "Euler-Lagrange correspondence of cellular automaton for traffic-flow models, J.Matsukidaira and KNishinari, Phys.Rev.Lett.Vol.90,No.8,p088701(2003)", we have found the transformation formula between Euler representation and Lagrange representation We have been able to obtain this formula by using a new algebraic formula between Max-Plus algebra and Step function. We have also applied this method to multi-value, multi-velocity traffic flow models and have been able to obtain Euler-Lagrange correspondence of the Fukui-Ishibashi model and the quick-start model Furthermore, we have applied this method to soliton cellular automata and have succeeded to find the Euler-Lagrange correspondence of Box and Ball system. We are now preparing the paper for this result.