 06134 Hakan L. Eliasson and Sergei B. Kuksin
 KAM for the nonlinear Schroedinger equation  a short presentation
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Apr 27, 06

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Abstract. We consider the $d$dimensional nonlinear Schr\"o\dinger
equation under periodic boundary conditions:
$$
i\dot u=\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u) ;\quad u=u(t,x),\;x\in\T^d
$$
where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is an analytic function with
$\hat V$ real and $F$ is a real analytic function in $\Re u$, $\Im u$ and $x$.
(This equation is a popular model for the `real' NLS
equation, where instead of the convolution term $V*u$ we have the
potential term $Vu$.) For $\ep=0$ the equation is linear and has
timequasiperiodic
solutions $u$,
$$
u(t,x)=\sum_{s\in \AA}\hat u_0(a)e^{i(a^2+\hat V(a))t}e^{i\sc{a,x}},
\quad 0<\hat u_0(a)\le1,
$$
where $\AA$ is any finite subset of $\Z^d$.
We shall treat $\omega_a=a^2+\hat V(a)$, $a\in\AA$, as free parameters
in some domain $U\subset\R^{\AA}$.
This is a Hamiltonian system in infinite degrees of freedom, degenerate
but with external parameters, and we shall describe
a KAMtheory which, in particular, will have the following consequence:
\smallskip
{\it If $\ep$ is sufficiently small, then there is a large subset
$U'$ of $U$ such that for all $\omega\in U'$
the solution
$u$ persists as a timequasiperiodic solution
which has all Lyapounov exponents
equal to zero and
whose linearized equation is reducible to constant
coefficients.}
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