  We deduce a generalized hydrodynamic limit for the box-ball system, which
explains how the densities of solitons 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 of
Ferrari, Nguyen, Rolla and Wang (cf. the original work of Takahashi and
Satsuma), 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.
