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.Comment: 20 pages, 7 figures, v2: final form to appear in J. Math. Sci. Univ. Toky

Integrability of Discrete Equations Modulo a Prime

We apply the 'almost good reduction' (AGR) criterion, which has been introduced in our previous (arXiv:1206.4456 and arXiv:1209.0223), to several classes of discrete integrable equations. We verify our conjecture that AGR plays the same role for maps of the plane define over simple finite fields as the notion of the singularity confinement does. We first prove that q-discrete analogues of the Painlev\'e III and IV equations have AGR. We next prove that the Hietarinta-Viallet equation, a non-integrable chaotic system also has AGR