Bambusi D.
LONG TIME STABILITY OF SOME SMALL AMPLITUDE SOLUTIONS
IN NONLINEAR SCHR\"ODINGER EQUATIONS
(252K, Postscript)
ABSTRACT. We consider small perturbations of the Zakharov-Shabat
nonlinear schroedinger equation on $[0,\pi]$ with vanishing or periodic
boundary conditions; we prove a Nekhoroshev type result for solutions
starting in the neighbourhood (in the $H^1$ topology) of the majority of
small amplitude finite dimensional invariant tori of the linearized
system. More precisely we will prove that along the considered solutions
all the actions of the linearized system are approximatively constant up
to times growing exponentially with the inverse of a suitable small
parameter.