Soliton Equation – Box-Ball System – Integrable Lattice Models

Box-ball system (BBS) is a discrete dynamical system which is expressed by movement of some kinds of balls in an infinite one dimensional array of boxes. Several years ago, the BBS is found to be obtained by so called the ul-tradiscrete limit of discrete KP equation (Hirota-Miwa equation). Using the solitonic properties (integrability) of KP hierarchy, the explicit soliton solu-tions and infinite number of conserved quantities of BBS have been obtained. Recently, another root to BBS was discovered: the time evolution patterns of BBS correspond to the configuration of an integrable lattice model at zero temperature with appropriate boundary conditions. In my talk, I would like to review the recent developments of BBS and give some new results for periodic BBS. References [1] T.Tokihiro, A.Nagai and J.Satsuma, ”Proof of solitonical nature of box and ball sys-tems by means of inverse ultra-discretization”, inverse Problems 15 (1999) 1639–1662