Arc lines for the box-ball system related to the ultradiscrete relativistic Toda equation

Abstract

The discrete relativistic Toda equation is derived from the compatibility condition of the spectral transformations of the Laurent biorthogonal polynomials. The ultradiscrete version of this equation is regarded as a time evolution of the soliton cellular automaton, a new kind of box-ball system. In the time evolution of this automaton, not only balls but also some boxes move. We introduce 10-arc lines for the automaton and obtain the conserved quantities corresponding to soliton sizes.