06-144 L. H. Eliasson and S. B. Kuksin

KAM for the non-linear Schroedinger equation (692K, pdf) May 2, 06

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Abstract. We consider the $d$-dimensional nonlinear Schr\"o\-dinger equation under periodic boundary conditions: $$ -i\dot u=\Delta u+V(x)*u+\ep|u|^2u;\quad u=u(t,x),\;x\in\T^d $$ 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.

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