The periodic box-ball systems (PBBS for short) are typical examples of soliton cellular automata. Each group of balls constitutes a soliton; the groups exhibit nonlinear scattering with each other. PBBS are not only a limit of classical soliton systems but also a limit of quantum integrable models. In particular, the limit of various types of transfer matrices of quantum integrable models provides different types of time evolutions ... , where ... . ... ... coincides with cyclic shift and the others form a commuting family of nontrivial time evolution operators that give integrability of the model. In this Demonstration, time evolutions proceed downwards. At the top is the rigged configuration, which is the action-angle variable (i.e., the constant of motion and the linearization parameter) corresponding to the bottom line.


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