We study an integrable vertex model with a periodic boundary condition
associated with U_q(A^{(1)}_n) at the crystallizing point q=0. It is an
(n+1)-state cellular automaton describing the factorized scattering of
solitons. The dynamics originates in the commuting family of fusion transfer
matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the
periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum
group, we develop an inverse scattering/spectral formalism and solve the
initial value problem based on several conjectures. The action-angle variables
are constructed representing the amplitudes and phases of solitons. By the
direct and inverse scattering maps, separation of variables into solitons is
achieved and nonlinear dynamics is transformed into a straight motion on a
tropical analogue of the Jacobi variety. We decompose the level set into
connected components under the commuting family of time evolutions and identify
each of them with the set of integer points on a torus. The weight multiplicity
formula derived from the q=0 Bethe equation acquires an elegant interpretation
as the volume of the phase space expressed by the size and multiplicity of
these tori. The dynamical period is determined as an explicit arithmetical
function of the n-tuple of Young diagrams specifying the level set. The inverse
map, i.e., tropical Jacobi inversion is expressed in terms of a tropical
Riemann theta function associated with the Bethe ansatz data. As an
application, time average of some local variable is calculated.