Soliton Decomposition of the Box-Ball System. (3rd September 2021)

• Record Type: Journal Article
• Authors: Ferrari, Pablo A.; Nguyen, Chi; Rolla, Leonardo T.; Wang, Minmin
• Is Part Of: Forum of mathematics. Volume 9(2021)
• Publication Date: 2021-09-03
• Subjects: 37B15 -- 60K35 -- 82C20 -- 82C70; box-ball systems -- solitons -- invariant measures; Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2021.49
• Languages: English
• ISSNs: 2050-5094

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 k th 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. 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.)