Box ball systems, soliton components and hydrodynamics

The box-ball system is a transport cellular automaton of balls on the integers introduced by Takahashi and Satsuma in 1990 as a discrete analogue of the KdV equation. The dynamics conserves solitons, solitary waves travelling at speed proportional to its size and conserve shape and speed even after colliding with other solitons. A ball configuration has soliton components that are fully conserved by the dynamics. The dynamic of each component is just a shift. 

A random initial ball configuration with translation invariant distribution and independent translation invariant components is invariant for the dynamics. The soliton decomposition of a product measure with density less than 1/2 has independent components, and those are explicitly described. In this case, the decomposition is related to the tree induced by the random walk with increments given by the ball configuration. 

In the translation invariant case, the asymptotic speeds of solitons are computed and shown to satisfy an universal system of linear equations corresponding to the Generalized Gibbs Ensemble of conservative systems.

The talk will require familiarity with basic concepts in stochastic processes. 

The talk is based on joint work with Chi Nguyen, Leo Rolla, Minmin Wang and Davide Gabrielli.  arXiv 1806.02798, 1812.02437 and 1906.06405.