Applicaticns of Ultradiscretization Method

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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.