Box-Ball system and Toda equation
A Box-Ball system (BBS) is constructed by a limiting procedure called ultra-discretization from the discrete KdV equation or the Toda lattice equation. In this talk, the relation between the BBS with the periodic boundary condition (PBBS) and discrete Toda equation. In particular, we show the relation between the conserved quantities of PBBS and a spectral curve of the Toda equation. Fundamental cycle of PBBS and initial value problem of PBBS are also discussed.