Soliton Decomposition of the Box-Ball System. (3rd September 2021) - Ferrari, Pablo A.; Nguyen, Chi; Rolla, Leonardo T.; Wang, Minmin
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 k th 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 $\frac {1}{2}$ . 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 an effective speed $v_k$ . The vector of speeds satisfies a system of linear equations related with the generalised Gibbs ensemble of conservative laws. Abstract: BBS dynamics for independent and identically distributed initial configuration with density 0.25. Time is going down. Straight red lines are deterministic and computed using Theorem 1.2 . (High resolution, color online.)