A KAM theorem for Hamiltonian networks with long ranged couplings

We consider Hamiltonian networks of long ranged and weakly coupled oscillators with variable frequencies. By deriving an abstract infinite dimensional KAM type of theorem, we show that for any given positive integer N and a fixed, positive measure set O of N variable frequencies, there is a subset O ∗ ⊂ O of positive measure such that each ω ∈ O ∗ corresponds to a small amplitude, quasi-periodic breather (i.e., a solution which is quasi-periodic in time and exponentially localized in space) of the Hamiltonian network with N-frequencies which are slightly deformed from ω.