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KAM tori in 1D random discrete nonlinear Schrödinger model?

M. Johansson, G.Kopidakis and S. Aubry


EPL 91, 50001 (2010)

(Highlighted as "editor's choice" by the journal)



The KAM (Kolmogorov-Arnold-Moser) theorem has fundamental implications for classical conservative many-body problems such as solar system dynamics. In simple terms, it states that most regular (quasiperiodic) trajectories are stable for any dynamical system, that can be considered as a weak perturbation of  some integrable system (having as many constants of motion as degrees of  freedom). The probability that a random initial condition will generate such a "KAM torus" goes to one when the perturbation goes to zero, and decreases for increasing perturbations. The remaining initial conditions become unstable due to resonances, and typically generate chaotic motion.
     KAM theory becomes more complex if the  number of degrees of freedom increases, due to more possible resonances, and very few extensions to infinite-dimensional systems are known. Here, we use empirical and numerical arguments to propose an extension to a physically important class of weakly  nonlinear lattices with strong disorder. Solutions to the unperturbed (linear) system are then exponentially (Anderson) localized in space, limiting the number of significant resonances.
     A main implication of our work, illustrated by the figure, is that such systems should exhibit two kinds of initial wave packets (intricately nested as a Cantor-like set): (i) those generating spatially localized, regular trajectories (KAM tori); and (ii) initially chaotic wave packets, spreading in space. This effect may be observable in disordered optical waveguide arrays, or with ultracold bosonic atoms.

Evolution of small perturbations to three nearby initial states: red and blue curves indicate stable, localized KAM tori (bounded perturbation); green curve in-between a chaotic, spatially spreading, trajectory (exponentially increasing perturbation).

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Last updated: 03/08/11