KAM for the Non-Linear Schrödinger Equation - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Annals of Mathematics Année : 2010

KAM for the Non-Linear Schrödinger Equation

Résumé

We consider the $d$-dimensional nonlinear Schrödinger 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 time--quasi-periodic solutions $u$, $$ u(t,x)=\sum_{a\in Å}\hat u(a)e^{i(|a|^2+\hat V(a))t}e^{i\sc{a,x}} \quad (|\hat u(a)|>0), $$ where $Å$ is any finite subset of $\Z^d$. We shall treat $\omega_a=|a|^2+\hat V(a)$, $a\inÅ$, as free parameters in some domain $U\subset\R^{Å}$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: 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.

Dates et versions

hal-00832755 , version 1 (11-06-2013)

Identifiants

Citer

Hakan Eliasson, Sergei Kuksin. KAM for the Non-Linear Schrödinger Equation. Annals of Mathematics, 2010, 172 (1), pp.371-435. ⟨10.4007/annals.2010.172.371⟩. ⟨hal-00832755⟩
134 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More