Dario BAMBUSI

Dipartimento di Matematica
Via Saldini 50
20133 Milano
 Italie
bambusi@mat.unimi.it

Nekhoroshev theorem for small amplitude solutions in some nonlinear PDE's

Abstract : Consider a perturbation of linear Hamiltonian system with the following properties: (i) the linear dynamics is periodic (ii) there exists a finite order Birkhoff normal form which is integrable and quasi convex as a function of the action variables. I will consider small amplitude initial data in which the majority of the energy is stored in a finite number of oscillators, and show that along the corresponding solutions all the actions of the linearized system are approximatively constant up to times growing exponentially with a suitable small

parameter. An application to the nonlinear Shr\"odinger equation on a segment will be given. In particular it will be proved that given initial data which are intere analytic functions of finite order then all the Fourier coefficients are approximate constants of motions for times longer than any power of a suitable small parameter.