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Ultradiscrete analogues of the nonautonomous discrete finite Toda lattice and its modified system are explicitly given. It is shown that these two systems can be seen as the box-ball system with size limits and one with speed limits, respectively. Particular solutions and the Lax forms of these ultradiscrete systems on the finite lattice are also discussed.

Box-ball systems related to the nonautonomous ultradiscrete Toda equation on the finite lattice