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 $\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.

Relationships between two linearizations of the box-ball system: Kerov–Kirillov–Reschetikhin bijection and slot configuration

“…This procedure turns out to be useful in order to establish connections between the KKR bijection and the seat number configuration; see Proposition 2.2 for details. In particular, we give an intuitive meaning to the KKR bijection, which was a purely combinatorial object by means of the carrier with seat numbers. As a result, we obtain an explicit relation between the KKR bijection and the slot configuration, an open problem addressed in [FNRW]. Our results reveal that the slot configuration can be defined independently of the notion of solitons; see Section 2.1 and Proposition 2.3 for details.…”

“…As a direct consequence of Theorem 2.2 and the result in [KOSTY] quoted as Theorem 4.1 in Section 4, we see that the slot configuration linearizes the BBS with finite capacity BBS(ℓ). More precisely, we obtain the following theorem, which is a generalization of [FNRW,Theorem 1.4] for the case ℓ < ∞.…”

“…In [CS], the GHD limit for the BBS with infinite carrier capacity (ℓ = ∞) is rigorously derived, and the use of the slot decomposition is crucial in their strategy of the proof. However, the assumption ℓ = ∞ is not necessary for most of the proof and is only needed to use [FNRW,Theorem 1.4], the linearization property of the slot decomposition. Therefore, combining Theorem 2.3 and the strategy in [CS], the GHD limit for the BBS(ℓ) can be also derived in a rigorous way.…”

“…A relation between the two linearizations was studied in [KS]. Recently, another linearization using the notion of the slot configuration and the k - slots has been introduced in [FNRW]. The latter linearization is known to be useful to study the randomized BBS and its generalized hydrodynamics [CS, FG, FNRW], where generalized hydrodynamics is a relatively new theory of hydrodynamics for integrable systems; see the review [D] for details.…”

“…Recently, another linearization using the notion of the slot configuration and the k - slots has been introduced in [FNRW]. The latter linearization is known to be useful to study the randomized BBS and its generalized hydrodynamics [CS, FG, FNRW], where generalized hydrodynamics is a relatively new theory of hydrodynamics for integrable systems; see the review [D] for details. The aim of this paper is to introduce a new algorithm which also linearizes the BBS dynamics and reveals relations between the KKR bijection and the slot configuration.…”