Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms
 A. Doelman,
 V. Rottschäfer
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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 wavenumberintervals of the two (weakly) unstable modes is small—as, for instance, can be the case in doublelayer 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 quasiperiodic patterns in the underlying basic system. Thus, the ‘Landau reduction’ to the nonlocal system has no significant influence on the stationary quasiperiodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to socalled ‘localised structures’ in the underlying system: They connect simple periodic patterns atx → ± ∞. 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.
 Title
 Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms
 Journal

Journal of Nonlinear Science
Volume 7, Issue 4 , pp 371409
 Cover Date
 199708
 DOI
 10.1007/BF02678142
 Print ISSN
 09388974
 Online ISSN
 14321467
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Authors

 A. Doelman ^{(1)}
 V. Rottschäfer ^{(1)}
 Author Affiliations

 1. Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508TA, Utrecht, The Netherlands