A finite Toda representation of the box-ball system with box capacity

TL;DR

This paper extends the connection between the finite ultradiscrete Toda lattice and the box-ball system to include variable box capacities and carrier capacities, introducing new rules and concepts for this generalized model.

Contribution

It introduces a novel finite Toda representation for the box-ball system with variable capacities, including new rules and an expansion map for the extended system.

Findings

Established a connection between the extended Toda lattice and the variable-capacity box-ball system

Introduced new carrier rules: size limit for solitons and recovery of balls

Presented a particular solution for a special case of the extended system

Abstract

A connection between the finite ultradiscrete Toda lattice and the box-ball system is extended to the case where each box has own capacity and a carrier has a capacity parameter depending on time. In order to consider this connection, new carrier rules "size limit for solitons" and "recovery of balls", and a concept "expansion map" are introduced. A particular solution to the extended system of a special case is also presented.