Soliton Decomposition of the Box-Ball System

Abstract

Abstract 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 kth component moves rigidly at speed k. Let $\zeta $ be a translation-invariant family of independent random vectors under a summability condition and $\eta $ be the ball configuration with components $\zeta $ . We show that the law of $\eta $ is translation invariant and invariant for the BBS. This recipe allows us to construct a large family of invariant measures, including product measures and stationary Markov chains with ball density less than $\frac {1}{2}$ . 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\to \infty $ to an effective speed $v_k$ . The vector of speeds satisfies a system of linear equations related with the generalised Gibbs ensemble of conservative laws.