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1997, English, Unpublished edition: Singularly Perturbed and Nonlocal Modulation Equations for Systems with Interacting Instability Mechanisms Doelman, A.; Rottschäfer, V.

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Physical Description
• application/​pdf
• Article /​ Letter to the editor
Published
• 1997
Language
• English

Edition details

Title
• Singularly Perturbed and Nonlocal Modulation Equations for Systems with Interacting Instability Mechanisms
Creator
• Doelman, A.
• Rottschäfer, V.
Published
• 1997
Physical Description
• application/​pdf
• Article /​ Letter to the editor
Subjects
Notes
• Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small---as, for instance, can be the case in double-layer convection. Based on these assumptions we first derive a singularly perturbed modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the Landau reduction' to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called localised structures' in the underlying system: They connect simple periodic patterns at $x \to \pm \infty$. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns.
Source
• Journal of Nonlinear Science (14321467) vol.7 (1997) nr.4 p.371
• It is not permitted to download or to forward/​distribute the text or part of it without the consent of the copyright holder (usually the author), other then for strictly personal, individual use.
Language
• English
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