Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms |
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Authors: | A Doelman V Rottschäfer |
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Institution: | (1) Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508TA Utrecht, The Netherlands |
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Abstract: | Summary 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 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. |
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