Packets of resonant modes in the Fermi-Pasta-Ulam system

We look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit. Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables we transform it into a sum of two parts in involution. At variance with the usual method of introducing normal modes, our constructive procedure allows us to exploit the complete resonance, while keeping the extensive nature of the system. Next we obtain a nonlinear approximation of an adiabatic invariant by considering a perturbation of the discrete nonlinear Schr\"odinger model. The fluctuations of this quantity are controlled via Gibbs measure estimates independent of te system size, for a large set of intial data at low specific energy. Finally, by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory, which fluctuation much smaller than expected according to standard statistical estimates.