Soliton decomposition of the Box-Ball System

Abstract

The Box-Ball System, shortly BBS, was introduced by Takahashi and Satsuma as a discrete counterpart of the KdV 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 $\eta$ 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 big 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 Generalized Gibbs Ensemble of conservative laws.

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Cited by in corpus (5)

• Dynamics of the box-ball system with random initial conditions via Pitman's transformation
• Randomized box-ball systems, limit shape of rigged configurations and Thermodynamic Bethe ansatz
• Current correlations, Drude weights and large deviations in a box-ball system
• Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings
• Dynamics of the ultra-discrete Toda lattice via Pitman's transformation