Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators

We prove existence and practical stability of breathers in chains of weakly coupled anharmonic oscillators. Precisely, for a large class of chains, we prove that there exist periodic solutions exponentially localized in space, with the property that, given an initial datum O(∈a) (with a ≥ 1/2) close to the phase space trajectory of the breather, then the corresponding solution remains at a distance O(∈a + √|t|exp(-∈-1/6)) from the above trajectory, up to times growing exponentially with the inverse of ∈, ∈ being a parameter measuring the size of the interaction among the particles. This result is deduced from a general normal form theorem for abstract Hamiltonian systems in Banach spaces, which we think could be interesting in itself.