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

We consider the nonlinear wave equation u(tt) - c(2)u(xx) = epsilon psi (u), u(t, 0) = u(t, pi) = 0, where psi is an analytic function satisfying psi (0) = 0 and psi' (0) not equal 0, For each value of the harmonic energy E (i.e. energy when epsilon = 0), We consider the closed curve gamma(E) which is the phase space trajectory of the linear mode with lowest frequency. if the perturbation parameter E and the harmonic energy E are small enough, then, near gamma(E) we construct, by means of a suitable "simplified equation", a closed curve gamma = gamma(E,epsilon) with the following property: provided the distance between the initial datum and such a curve is small enough, the corresponding solution remains close to gamma(E,epsilon) up to times exponentially long with 1/epsilon; one cannot expect the curves Y-E,Y-epsilon to be trajectories of solutions of (*). Such curves depend smoothly on the parameters E and epsilon. The above result is deduced from a general theorem on generic Hamiltonian perturbations of completely resonant Linear systems. (C) 1998 Elsevier Science B.V.