Box and ball system with numbered boxes and balls (Mathematical structures of integrable systems, their developments and applications)

Abstract

Box and ball systems (BBSs) are known as discrete dynamical systems in which motions of balls among successive infinite boxes are governed by an ultradiscrete integrable system. The equation of motion in the simplest BBS is the ultradiscrete version of the discrete Toda equation, which is one of famous discrete integrable systems. The discrete Toda equation is extended to two types of discrete hungry Toda (dhToda) equations, and their ultradiscretizations are shown to be the equations of motion in the BBSs in which either boxes or balls are numbered. In this paper, we propose a new box-ball system in which both boxes and balls are numbered, and show that its equation of motion is the ultradiscretization of a variant of the dhToda equations. With the help of a combinatorial technique, we describe conserved quantities of our new numbered BBS (nBBS). We also clarify its relationship to the hungry ε-BBS, which is derived from the ultradiscretization of another extension of the discrete Toda equation.