CHAOTIC BEHAVIOUR AND LOCALIZATION IN LINEAR AND NON LINEAR MEDIA

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CHAOTIC BEHAVIOUR AND LOCALIZATION IN LINEAR AND NON LINEAR MEDIA

J. Coste, J. Peyraud

To cite this version:

J. Coste, J. Peyraud. CHAOTIC BEHAVIOUR AND LOCALIZATION IN LINEAR AND NON LINEAR MEDIA. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-85-C3-85.

�10.1051/jphyscol:1989312�. �jpa-00229453�

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JOURNAL DE PHYSIQUE

Colloque C3, suppli?ment.au n 0 3 , Tome 50, mars 1989

CHAOTIC BEHAVIOUR AND LOCALIZATION IN LINEAR AND NON LINEAR MEDIA

J. COSTE and J. PEYRAUD

Laboratoire de Physique de la Matiere condensbe, CNRS UA-190, Universit6 de Nice. Parc Valrose, F-06034 Nice Cedex, France

This paper deals w i t h some aspects of localization and chaos i n linear quas,i-periodic media and periodic non linear media.

1

-

Linear auasi -oeriodic media.

The quasi-periodic media we have studied present refractive indexes w i t h b l i k e jumps on quasi-periodically located sites. Two model of quasi-periodic media have been considered, namely :

n= 1 +I,,, G(x-mp-nq) (p/q+o : golden mean) : Model I.

n= 1 +Zj G(x- jp/q mod( 1 )) (p/q-ta: golden mean) : Model I I We have shown that there exists a slow variable allowing a mathematical description of the propagation. We obtain striking results : i ) In the model I, where we find quasi-localized states w i t h a critical definition i n energy, ii)

In

the second model, where we obtain intermittency and a new phenomenon of "noise localization".

2- Non linear ~ e r i o d i c media.

We have both studied the light propagation on a discret model : n = 1 +PmcmS (x-m)

C , , , = E ( I + ~ / ~ ~ ~ ~ )

waveamplit amplitude)

and on a continuous model : 2 2

is,

+k ( I ⁺p,lrlZii 1 ⁺E COS(~X)I y = 0

We deal here w i t h non integrable dynamic systems These systems prove t o be equivalent to the study of a non linear Poincare mapping. The main results we have obtained are :

-There exist four types of localized solutions which correspond to the four Arnold strong resonances of the dynamical system. One of them i s the so-called "Gap soliton" already described by Mills and Trullinger.

-These solutions are weakly chaotic and the associated stochasticity i s responsible for physical effects f i t bounds the maximal spatial extension to the localized structure).

Finally, we have made a dynamical study of gap solitons. We have shown that these structures are usually unstable giving rise to solitary propagative waves.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989312