A property of exponential stability in nonlinear wave equations near the fundamental linear mode

We consider the nonlinear wave equation Utt - C2Uxx = εψ(u), u(t,0) = u(t,π) = 0, (*) where ψ is an analytic function satisfying ψ(0) = 0 and ψ'(0) ≠ 0. For each value of the harmonic energy E (i.e. energy when ε = 0), we consider the closed curve which is the phase space trajectory of the linear mode with lowest frequency. If the perturbation parameter ε and the harmonic energy E are small enough, then, near γE we construct, by means of a suitable "simplified equation", a closed curve γ= γE,ε with the following property: provided the distance between the initial datum and such a curve is small enough, the corresponding solution remains close to γE,ε up to times exponentially long with 1/ε; one cannot expect the curves γE,ε to be trajectories of solutions of (*). Such curves depend smoothly on the parameters E and ε. The above result is deduced from a general theorem on generic Hamiltonian perturbations of completely resonant linear systems.