We study the equilibration properties of classical integrable field theories at a finite energy density, with a time evolution that starts from initial conditions far from equilibrium. These classical field theories may be regarded as quantum field theories in the regime of high occupation numbers. This observation permits to recover the classical quantities from the quantum ones by taking a proper $\hbar \rightarrow 0$ limit. In particular, the time averages of the classical theories can be expressed in terms of a suitable version of the LeClair-Mussardo formula relative to the Generalized Gibbs Ensemble. For the purposes of handling time averages, our approach provides a solution of the problem of the {\em infinite gap solutions} of the Inverse Scattering Method.

67 pages (+ 9 pages of appendices), 26 figures; review article for special issue of JSTAT on Quantum Integrability in Out of Equilibrium Systems; comments are welcome