Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization

Abstract

A soliton cellular automaton, which represents movement of a finite number of balls in an array of boxes, is investigated. Its dynamics is described by an ultra-discrete equation obtained from an extended Toda molecule equation. The rules for soliton interactions and factorization property of the scattering matrices (Yang-Baxter relation) are proved by means of inverse ultra-discretization. The conserved quantities are also presented and used for another proof of the solitonical nature.