On Nekhoroshev estimates for a nonlinear Schrodlinger equation and a theorem by Bambusi

We consider the nonlinear Schrodinger equation iu(t) = u(xx) - mu - f(/u/(2))u on a finite x-interval with Dirichlet boundary conditions. Assuming that f is real analytic with f(0) = 0 and f'(0) not equal 0, we show that the equilibrium solution u equivalent to 0 enjoys a certain kind of Nekhoroshev stability. If most of the energy is located in finitely many modes and sufficiently small, then the amplitudes of these modes are almost constant over a time interval,which is exponentially long in the inverse of the total energy. This result is due to Bambusi, but the proof given here is conceptually and technically simpler. It may also apply to a larger class of nonlinear partial differential equations.