Renormalization group and the Melnikov problem for PDE's

We give anew proof of persistence of quasi-periodic, low dimensional ellipt ic tori in infinite dimensional systems. The proof is based on a renormaliz ation group iteration that was developed recently in [BGK] to address the s tandard KAM problem, namely, persistence of invariant tori of maximal dimen sion in finite dimensional, near integrable systems. Our result covers situ ations in which the so called normal frequencies are multiple. In particula r, it provides a new proof of the existence of small-amplitude, quasi-perio dic solutions of nonlinear wave equations with periodic boundary conditions .