Soliton decomposition of the Box-Ball System

PABLO A. FERRARI; CHI NGUYEN; LEONARDO T ROLLA; MINMIN WANG

Forum of Mathematics, Sigma

Cambridge University

Lugar: Cambridge; Año: 2021 vol. 9 p. 1 - 37

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.