Branching of Cantor manifolds of elliptic tori and applications to PDEs

We consider infinite dimensional Hamiltonian systems. First we prove the existence of “Cantor manifolds” of elliptic tori -of any finite higher dimensionaccumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are “branching” points of other Cantor manifolds of higher dimensional tori. We also provide a positive answer to a conjecture of Bourgain [8] proving the existence of invariant elliptic KAM tori with tangential frequency constrained to a fixed Diophantine direction. These results are obtained under the natural nonresonance and nondegeneracy conditions. As applications we prove the existence of new kinds of quasi periodic solutions of the one dimensional nonlinear wave equation. The proofs are based on averaging normal forms and a sharp KAM theorem, whose advantages are an explicit characterisation of the Cantor set of parameters, quite convenient for measure estimates, and weaker smallness conditions on the perturbation.