Another generalization of the box–ball system with many kinds of balls

Abstract

A cellular automaton that is a generalization of the box-ball system with either many kinds of balls or finite carrier capacity is proposed and studied through two discrete integrable systems: nonautonomous discrete KP lattice and nonautonomous discrete two-dimensional Toda lattice. Applying reduction technique and ultradiscretization procedure to these discrete systems, we derive two types of time evolution equations of the proposed cellular automaton, and particular solutions to the ultradiscrete equations.

Nonautonomous discrete elementary Toda orbits and their ultradiscretization

“…We define q (k,t) i = c i and conclude the proof of (18). The proof of ( 20) is performed similarly (18).…”

“…Equations ( 4) and (5) are called the ultradiscrete Toda lattice. Various extensions of the BBS can be obtained from the ultradiscretization of discrete integrable systems [17,18,27]. Interestingly, the BBS can be obtained not only by the limit of classical integrable system as seen above, but also by the limit of a solvable lattice model (see [12] for detailed exposition).…”