Soliton decomposition of the Box-Ball System

The box-ball system (BBS) was introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg-de Vries equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size k solitons in each k-slot. The dynamics of the components is linear: the k-th component moves rigidly at speed k. Let ? be a translation invariant family of independent random vectors under a summability condition and ? the ball configuration with components ?. We show that the law of ? is translation invariant and invariant for the BBS. This recipe allows us to construct a big family of invariant measures, including product measures and stationary Markov chains with ball density less than 1/2. We also show that starting BBS with an ergodic measure, the position of a tagged k-soliton at time t, divided by t converges as t ? 8 to an effective speed vk. The vector of speeds satisfies a system of linear equations related with the Generalized Gibbs Ensemble of conservative laws.