Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Soliton decomposition of the Box-Ball System

Pablo A. Ferrari, Chi Nguyen, Leonardo T. Rolla, Minmin Wang

TL;DR

We study the Box-Ball System in the density regime $\\lambda<\\frac{1}{2}$ and introduce a slot decomposition that maps a configuration to independent soliton components. Each component evolves linearly under the Box-Ball dynamics, enabling a reconstruction of the full configuration from the components. We construct a broad family of $T$-invariant measures via Palm theory, including product measures and stationary Markov chains, and we derive almost sure asymptotic speeds $(v_k)$ for $k$-solitons under shift-ergodic initial states, governed by a linear system tied to soliton densities. Our framework connects to ideas from generalized hydrodynamics and the generalized Gibbs ensemble, and yields explicit recursive formulas for speeds in the independent-component case.

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 $ζ$ 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 $\frac12$. 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.

Soliton decomposition of the Box-Ball System

TL;DR

We study the Box-Ball System in the density regime and introduce a slot decomposition that maps a configuration to independent soliton components. Each component evolves linearly under the Box-Ball dynamics, enabling a reconstruction of the full configuration from the components. We construct a broad family of -invariant measures via Palm theory, including product measures and stationary Markov chains, and we derive almost sure asymptotic speeds for -solitons under shift-ergodic initial states, governed by a linear system tied to soliton densities. Our framework connects to ideas from generalized hydrodynamics and the generalized Gibbs ensemble, and yields explicit recursive formulas for speeds in the independent-component case.

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 solitons in each -slot. The dynamics of the components is linear: the -th component moves rigidly at speed . Let be a translation invariant family of independent random vectors under a summability condition and the ball configuration with components . We show that the law of 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 . We also show that starting BBS with an ergodic measure, the position of a tagged -soliton at time , divided by converges as to an effective speed . The vector of speeds satisfies a system of linear equations related with the Generalized Gibbs Ensemble of conservative laws.

Paper Structure

This paper contains 22 sections, 19 theorems, 108 equations, 14 figures, 3 algorithms.

Table of Contents

  1. Preliminaries and results
  2. Identifying and tracking solitons
  3. Soliton nesting and motion
  4. Asymptotic speeds
  5. Slot decomposition
  6. Invariant measures
  7. Slot decomposition
  8. Slots and components
  9. Reconstructing the configuration from the components
  10. Evolution of components
  11. Invariant measures
  12. Construction of the measures
  13. Invariance of the reconstructed configuration
  14. Expected excursion length
  15. Palm transformations
  16. ...and 7 more sections

Key Result

Proposition 1.4

For any $\eta\in\mathcal{X}$ and $A \subseteq \mathbb{Z}$, there is a $k$-soliton $\gamma\in\Gamma_k\eta$ with tail $\mathcal{T}(\gamma)=A$ if and only if there is a $k$-soliton $\gamma^1 \in \Gamma_k(T\eta)$ with head $\mathcal{H}(\gamma^1) = A$.

Figures (14)

  • Figure 1.1: Applying the Takahashi--Satsuma algorithm to a sample configuration. Dots represent records. On the left we have the resulting word after successive iterations. Identified solitons are shown in bold once and then with a color corresponding to their size. The algorithm is applied to each excursion separately, so the rightmost $1$-soliton in the picture is ignored by this instance of the procedure. (color online)
  • Figure 1.2: Here we show $I(\gamma)$ in an example with 9 records, a 5-soliton, a 4-soliton, two 3-solitons, two 2-solitons and two 1-solitons, with one color for each size. In this example, a 1-soliton is contained in a 2-soliton, both 2-solitons are contained in the 5-soliton, both 3-solitons are contained in the 4-soliton. $I(\gamma)$ is underlined with the same color as $\gamma$, and black zeros are records. (color online)
  • Figure 1.3: Simulation for an i.i.d. configuration with density $0.15$. The transparent red lines have deterministic slopes computed by Theorem \ref{['thm:speedsexplicit']}, which have been manually shifted so that they would overlay a soliton. This window covers 2000 sites and 140 time steps going downwards, and has been stretched vertically by a factor of $5$. The figure in the first page is the same except for the density. (high resolution, color online)
  • Figure 1.4: Simulation for $(\rho_k)_k=(.006,.005 ,.1,.003,0,0,0,\dots)$. The initial configuration was obtained by first appending one $k$-soliton with probability $\rho_k$ after each record, and then applying $T$ a number of times in order to mix. As in Fig. \ref{['fig:speeds1']} it is a 2000x140 window stretched by 5, and red lines are deterministic. (high resolution, color online)
  • Figure 2.1: Time-evolution of a walk under seven iterations of $T$. This example has four solitons, of size 7, 5, 3 and 1. Different colors are used to highlight their conservation. To facilitate view we have shifted the walk at time $t$ by $t$ units down. (color online)
  • ...and 9 more figures

Theorems & Definitions (38)

  • Proposition 1.4
  • Lemma 1.6
  • Lemma 1.7
  • Lemma 1.8
  • Theorem 1.1
  • Proposition 1.12
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • ...and 28 more