  Using generalized hydrodynamics (GHD), we develop the Euler hydrodynamics of
classical integrable field theory. Classical field GHD is based on a known
formalism for Gibbs ensembles of classical fields, that resembles the
thermodynamic Bethe ansatz of quantum models, which we extend to generalized
Gibbs ensembles (GGEs). In general, GHD must take into account both solitonic
and radiative modes of classical fields. We observe that the quasi-particle
formulation of GHD remains valid for radiative modes, even though these do not
display particle-like properties in their precise dynamics. We point out that
because of a UV catastrophe similar to that of black body radiation, radiative
modes suffer from divergences that restrict the set of finite-average
observables; this set is larger for GGEs with higher conserved charges. We
concentrate on the sinh-Gordon model, which only has radiative modes, and study
transport in the domain-wall initial problem as well as Euler-scale
correlations in GGEs. We confirm a variety of exact GHD predictions, including
those coming from hydrodynamic projection theory, by comparing with Metropolis
numerical evaluations.
