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

TL;DR

This paper introduces a generalized cellular automaton extending the box-ball system to multiple ball types and finite carrier capacity, analyzed via integrable discrete systems and ultradiscretization.

Contribution

It proposes a new generalized cellular automaton model and derives its evolution equations using advanced integrable systems techniques.

Findings

Derived two types of time evolution equations for the automaton.

Obtained particular solutions to the ultradiscrete equations.

Connected the automaton's behavior to nonautonomous discrete KP and Toda lattices.

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.