Soliton decomposition of the Box-Ball System

Ferrari, Pablo A, Nguyen, Chi, Rolla, Leonardo T and Wang, Minmin (2021) Soliton decomposition of the Box-Ball System. Forum of mathematics, Sigma, 9. a60 1-37. ISSN 2050-5094

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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 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 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 → ∞ to an effective speed vk. The vector of speeds satisfies a system of linear equations related with the Generalized Gibbs Ensemble of conservative laws.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
SWORD Depositor: Mx Elements Account
Depositing User: Mx Elements Account
Date Deposited: 06 Jul 2021 06:56
Last Modified: 07 Sep 2021 10:30
URI: http://sro.sussex.ac.uk/id/eprint/100152

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