A Kam Theorem for Space-Multidimensional Hamiltonian PDE

We present an abstract KAM theorem, adapted to space-multidimensional hamiltonian PDEs with smoothing non-linearities. The main novelties of this theorem are that: $\bullet$ the integrable part of the hamiltonian may contain a hyperbolic part and as a consequence the constructed invariant tori may be unstable. $\bullet$ It applies to singular perturbation problem. In this paper we state the KAM-theorem and comment on it, give the main ingredients of the proof, and present three applications of the theorem .