L. H. Eliasson and S. B. Kuksin
KAM for the non-linear Schroedinger equation
(692K, pdf)

ABSTRACT.  We consider the $d$-dimensional nonlinear Schr\"o\-dinger 
equation under periodic boundary conditions: 
$$ 
 -i\dot u=\Delta u+V(x)*u+\ep|u|^2u;\quad u=u(t,x),\;x\in\T^d 
$$ 
where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is an analytic function with 
$\hat V$ real. (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 
time--quasi-periodic 
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 KAM-theory 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 time--quasi-periodic solution 
which has all Lyapounov exponents 
equal to zero and 
whose linearized equation is reducible to constant 
coefficients.