The box-ball system is a one-dimensional transport cellular automaton that exhibits soliton behavior, analogous to soliton solutions of the KdV equation. A ball configuration has soliton components travelling ballistically. This linearization yields generalized hydrodynamic theorems and a huge family of invariant measures. We will describe the decomposition and the generalized hydrodynamic limit, a system of partial differential equations indexed by the soliton velocities.  In the continuous case, the decomposition of continuous random configurations associated to the so-called zig-zag random walk gives a Poisson process. Based on works with Inés Armendáriz, Pablo Blanc, Davide Gabrielli, Chi Nguyen, Leo Rolla and Minmin Wang.