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Mathematical analysis of sideband instabilities with application to rayleigh-bénard convection

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Summary

We introduce a new method for the analysis of sideband instabilities which are important for periodic patterns appearing in systems close to the instability threshold. The method relies on a two-fold application of the Liapunov-Schmidt reduction procedure, a first application to the nonlinear bifurcation problem and a second application to the linear spectral problem. We obtain rigorous results on the spectrum of the associated linearization in spaces allowing for general sideband perturbations by treating the sideband vector and the spectral parameter as small bifurcation parameters.

We apply the theory to the small roll solutions in the Rayleigh-Bénard convection and derive domains in Rayleigh, Prandtl, and wave number space where the rolls are unstable. We recover the Eckhaus, zigzag, and skew-varicose instabilities obtained earlier by formal methods.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Hannover, W3000, Hannover 1, Germany

    A. Mielke

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  1. A. Mielke
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Additional information

Communicated by Jerrold Marsden and Stephen Wiggins

This paper is dedicated to the memory of Juan C. Simo

This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.

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Mielke, A. Mathematical analysis of sideband instabilities with application to rayleigh-bénard convection. J Nonlinear Sci 7, 57–99 (1997). https://doi.org/10.1007/BF02679126

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  • DOI: https://doi.org/10.1007/BF02679126

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