Soliton decomposition of the Box-Ball System

Abstract

The Box-Ball System 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. Conservation of solitons suggests that this dynamics has many spatially-ergodic invariant measures besides the i.i.d. distribution. Meanwhile, solitons of different sizes interact through a momentary change of speeds during collision, which cumulatively affects their asymptotic speeds, suggesting that the speeds are determined by such interaction. In order to understand general invariant measures and soliton interactions, we introduce a decomposition of configurations through slots, reducing the dynamics to a simple hierarchical translation of different components. Using this property we obtain an explicit recipe to construct a rich family of invariant measures. Finally, we obtain explicit equations for the soliton speeds in terms of spacial density of solitons. BBS dynamics for i.i.d. initial configuration with density 0.25. Straight red lines are deterministic and computed using Theorem 1.2.

BBS invariant measures with independent soliton components

“…The independence of components combined with Theorem 11 imply that π λ and the Ising-like measures are invariant for BBS. These facts were proven directly by Croydon, Kato, Sasada and Tsujimoto [1], using reversibility of the carrier process illustrated in (1); see also [3].…”

“…Soliton decomposition of ball configurations [3] We say that a k-soliton γ is appended to k-slot j of η if its support is strictly included in the open integer interval with extremes in the k-slots j and j + 1:…”

“…We have that E is in bijection with S so that a slot diagram completely codifies an excursion. We now construct the map ε → x[ε] and its inverse x → ε[x] (see [3] and [2] for more details).…”

“…A configuration of balls can be mapped to a walk that jumps one unit up at occupied boxes and one unit down at empty boxes [1] [3]. The excursions of the walk are the pieces of configuration between two consecutive down records.…”

Hydrodynamic Equations for the Toda Lattice

“…The dynamics is given through a deterministic map iterated many times. The hydrodynamics of the box-ball system has been studied in considerable detail [34,65,96,98].…”