Abstract
Ultradiscretization (UD) is a limit operation which transforms a smooth equation into a piecewise linear equation. Since time evolution of a cellular automaton (CA) is described by a piecewise linear equation, UD provides specific CAs which inherit important properties from the equations of interest, in particular, integrable equations. A Box-Ball system (BBS) is regarded as a CA obtained from a soliton equation through UD. It has soliton solutions, infinite number of conserved quantities, and relation with quantum integrability. UD for other equations such as Painlevé equations has been studied and interesting CAs or piecewise linear equations were obtained.
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