Generalized hydrodynamic limit for the box-ball system

2021/01/22 Fri 15:00 - 16:30
David Croydon

I will talk about a recent joint work with Makiko Sasada (University of Tokyo) in which we obtain a generalized hydrodynamic limit for the box–ball system of Takahashi and Satsuma. This explains how the densities of solitons (i.e. solitary waves) of different sizes evolve asymptotically under Euler space-time scaling. To describe the limiting soliton flow, we introduce a continuous state space analogue of the soliton decomposition, namely we relate the densities of solitons of given sizes in space to corresponding densities on a scale of ‘effective distances’, where the dynamics are linear. For smooth initial conditions, we further show that the resulting evolution of the soliton densities in space can alternatively be characterised by a partial differential equation, which naturally links the time-derivatives of the soliton densities and the ‘effective speeds’ of solitons locally. Although our arguments and main result are essentially deterministic, they are inspired by work of Ferrari, Nguyen, Rolla and Wang (and Ferrari and Gabrielli) for stationary random configurations, which I also plan to describe.