Cellular automata and ultra-discrete Painlevé equations. (English) Zbl 0912.34005
Summary: Starting from integrable cellular automata the authors present a novel form of Painlevé equations. These equations are discrete in both the independent variable and the dependent one. They capture the essence of the behavior of the Painlevé equations, organize themselves into a coalescence cascade and possess special solutions. A necessary condition for the integrability of cellular automata is presented. The authors conclude with a discussion of the notion of integrability of cellular automata under examination.
MSC:
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
| 68Q80 | Cellular automata (computational aspects) |
| 34A26 | Geometric methods in ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
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