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We study an integrable vertex model with a periodic boundary conditionassociated with U_q(A_n^{(1)}) at the crystallizing point q=0. It is an(n+1)-state cellular automaton describing the factorized scattering ofsolitons. The dynamics originates in the commuting family of fusion transfermatrices and generalizes the ultradiscrete Toda/KP flow corresponding to theperiodic box-ball system. Combining Bethe ansatz and crystal theory in quantumgroup, we develop an inverse scattering/spectral formalism and solve theinitial value problem based on several conjectures. The action-angle variablesare constructed representing the amplitudes and phases of solitons. By thedirect and inverse scattering maps, separation of variables into solitons isachieved and nonlinear dynamics is transformed into a straight motion on atropical analogue of the Jacobi variety. We decompose the level set intoconnected components under the commuting family of time evolutions and identifyeach of them with the set of integer points on a torus. The weight multiplicityformula derived from the q=0 Bethe equation acquires an elegant interpretationas the volume of the phase space expressed by the size and multiplicity ofthese tori. The dynamical period is determined as an explicit arithmeticalfunction of the n-tuple of Young diagrams specifying the level set. The inversemap, i.e., tropical Jacobi inversion is expressed in terms of a tropicalRiemann theta function associated with the Bethe ansatz data. As anapplication, time average of some local variable is calculated.

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for A^{(1)}_n