Applicaticns of Ultradiscretization Method

超离散化是利用经验型非解析极限对差分方程的因变量进行离散化的一种方法。该方法与微分相结合，实现了由微分经差分到超离散的一系列离散过程。我们将这些方法应用于各种方程，并试图在本研究中找到它们的应用。首先，我们通过对著名的Burgers方程进行超离散化，成功地得到了一个交通阻塞过程的模型。对Burgers方程的超离散模拟是一个规则-184元胞自动机。其次，我们提出了一个新的数字方程，可以显示模式的形成过程。该数字方程由超离散化方法的一个基础max-plus代数给出。最后，对一类二阶非线性可积差分方程进行了超离散化，并分析了它们的解结构。超离散解类似于mod函数。我们从动力系统理论的角度澄清了解的行为，得到了显式解。

Ultradiscretization is a method to discretize a dependent variable of difference equation by using an non-analytic limit of exp type. Combination of this method and differentiation enables a series of discretizing process from differential to ultradiscrete via difference. We apply these methods to various equations and try to find their application in this research.First, we succeeded to obtain a model which show a traffic jamming process by ultradiscretizing the well-known Burgers equation. The ultradiscrete analogue to Burgers equation is a rule-184 cellular automaton.Next, we proposed a new digital equation which can show a pattern formation process. This digital equation is given by the max-plus algebra which is a base of the ultradiscretization method.Finally, we ultradiscretized some nonlinear integrable differenece equations of second order and analyzed their solution structure. Ultradiscrete solutions are similar to the mod function. We clarified the behavior of the solutions from a view point of the dynamical system theory and obtained explicit solutions.