An averaging theorem for quasilinear Hamiltonian PDEs

We study the dynamics of Hamiltonian quasilinear PDEs close to elliptic equilibria. Under a suitable nonresonance condition we prove an averaging theorem according to which any solution corresponding to smooth initial data with small amplitude remains very close to a torus up to long times. An application to quasilinear wave equations in an n-dimensional paralleliped is given.