The Bethe ansatz in a periodic box-ball system and the ultradiscrete   Riemann theta function
  Vertex models with quantum group symmetry give rise to integrable cellular automata at q=0. We study a prototype example known as the periodic box-ball system. The initial value problem is solved in terms of an ultradiscrete analogue of the Riemann theta function whose period matrix originates in the Bethe ansatz at q=0. 
Atsuo Kuniba and Reiho Sakamoto