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

A connection between the finite ultradiscrete Toda lattice and the box–ball system is extended to the case where each box has its 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.