The Kuramoto-Sivashinsky Equation: A Progress Report
Instabilities which develop in continuous media often lead to the formation of cellular structures periodic in space and/or time. One of the most important parameters which control the transition to turbulence in such systems is the aspect ratio. This quantity can be defined as the ratio of the lateral extension of the experimental enclosure to the typical size of the cells generated by the instability mechanism. When this ratio is small, confinement effects are strong. The situation is then best described by a small number of interacting modes and the theory of dissipative dynamical systems applies in a more or less straightforward way. The opposite limit of large aspect ratio has been of much concern recently. As explained by A. Newell in his lecture, the most important features of the dynamics of these structures close to onset are related to long wavelength low frequency spatiotemporal modulations. The dynamics of these modulations can be accounted for by envelope equations, the envelope being generically a complex function, slowly varying at the scale of the individual cells. In the most general approach, one seeks the envelope equation in the laterally infinite case by an expansion formalism involving both the envelope modulus and the inverse of the modulation length scales as small parameters. Lateral boundary conditions are an obvious source of modulation at a well defined lengthscale, the aspect ratio.
KeywordsLyapunov Exponent Bifurcation Diagram Unstable Mode Inertial Manifold Stability Window
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