KAM tori in 1D random discrete nonlinear Schrödinger model?

¹ Department of Physics, Chemistry and Biology (IFM), Linköping University - SE-581 83 Linköping, Sweden, EU
² Department of Materials Science and Technology, University of Crete - GR-71003 Heraklion, Greece, EU
³ Laboratoire Léon Brillouin, CEA Saclay - F-91191 Gif-sur-Yvette, France, EU
⁴ Max Planck Institute for the Physics of Complex Systems - Nöthnitzer Str. 38, D-01187 Dresden, Germany, EU

^a mjn@ifm.liu.se

Received: 9 July 2010
Accepted: 17 August 2010

Abstract

We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the random discrete nonlinear Schrödinger equation, appearing with a finite probability for a given initial condition with sufficiently small norm. Numerical support for the existence of a fat Cantor set of initial conditions generating almost periodic oscillations is obtained by analyzing i) sets of recurrent trajectories over successively larger time scales, and ii) finite-time Lyapunov exponents. The norm region where such KAM-like tori may exist shrinks to zero when the disorder strength goes to zero and the localization length diverges.