The generalized periodic ultradiscrete KdV equation and its background solutions

We investigate the ultradiscrete KdV equation with periodic boundary conditions where the two parameters (capacity of the boxes and that of the carrier) are arbitrary integers. We give a criterion to allow a periodic boundary condition when initial states take arbitrary integer values. Conserved quantities are constructed for the periodic systems. Construction of background solutions of the periodic ultradiscrete KdV equation from the Jacobi theta function is also presented.