 98153 Bambusi D.
 Nekhoroshev theorem for small amplitude solutions in
nonlinear Schr\"odinger equations
(729K, PS)
Mar 5, 98

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Abstract. We prove a Nekhoroshev type result\upccite{nek71}{nek77} for
the nonlinear \schro\ equation
$$
iu_t=u_{xx}muu \per (u^2)\ ,\autoeqno{1}
$$
with vanishing or periodic boundary conditions on $[0,\pi]$; here
$m\in\Re$ is a parameter and $\per :\Re\to\Re$ is a function analytic
in a neighborhood of the origin and such that $\per(0)=0$,
$\per'(0)\not=0$. More precisely, we consider the Cauchy problem for
\eqref{1} with initial data which extend to analytic integer functions
of finite order, and prove that all the actions of the linearized
system are approximate constants of motion up to times growing faster
than any negative power of the size of the initial datum. The proof is
obtained by a method which applies to Hamiltonian perturbations of
linear PDE's 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. Eq.~\eqref{1} satisfies (i) and (ii) when restricted to a
level surface of $\norma u_{L^2}$, which is an integral of motion. The
main technical tool used in the proof is a normal form lemma for
systems with symmetry which is also proved here.
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