“…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. We highlight that our results contribute to the growing literature concerning the randomization of the BBS [1,2,3,8,9,10,13,14,15,17,18], and that it also connects to current work on generalized hydrodynamics [4,16,19].…”