Soliton Decomposition of the Box-Ball System

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 ζ be a translation-invariant family of independent random vectors under a summability condition and η be the ball configuration with components ζ. We show that the law of η 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 ½ . 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 an effective speed vk . The vector of speeds satisfies a system of linear equations related with the generalised Gibbs ensemble of conservative laws.