Normal form and exponential stability for some nonlinear string equations

We study small amplitude solutions of the nonlinear wave equation utt - ucursive Greek chicursive Greek chi = ψ(u) u(0, t) = 0 = u(π, t) (0.1) with an analytic nonlinearity of the type ψ(u) = ±u2k-1 + script O sign(u2k) , k ≥ 2 . For this system we introduce a new method to compute the resonant normal form obtaining a simple formula for it. Then we specialize to the case k = 2 and find all periodic solutions of the averaged system and also of what we call the "simplified system", namely a Hamiltonian system that we will prove to approximate 0.1 up to an error exponentially small with the energy of the initial datum. Furthermore, for one of such orbits we prove a strong (Nekhoroshev type) stability property with respect to the complete dynamics.