KAM for the nonlinear Schrödinger equation


We consider the $d$-dimensional nonlinear Schrödinger equation under periodic boundary conditions: \[ -i\dot u=-\Delta u+V(x)*u+\varepsilon \frac{\partial F}{\partial \bar u}(x,u,\bar u), \quad u=u(t,x),\;x\in\mathbb{T}^d \] where $V(x)=\sum \hat{V}(a)e^{i\langle a,x\rangle}$ 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 $\varepsilon=0$ the equation is linear and has time–quasi-periodic solutions \[ u(t,x)=\sum_{a\in \mathcal{A}}\hat u(a)e^{i(|a|^2+\hat{V}(a))t}e^{i\langle a,x\rangle}, \quad |\hat u(a)|>0, \] where $\mathcal{A}$ is any finite subset of $\mathbb{Z}^d$. We shall treat $\omega_a=|a|^2+\hat V(a)$, $a\in\mathcal{A}$, as free parameters in some domain $U\subset\mathbb{R}^{\mathcal{A}}$.

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 $|\varepsilon|$ 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.


L. Hakan Eliasson

University of Paris 7, Department of Mathematics, Case 7052, 2 place Jussieu, Paris, France

Sergei B. Kuksin

CMLS, Ecole Polytechnique, 91128 Palaiseau, France