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Instabilities and Dynamics of Weakly Subcritical Patterns

Published online by Cambridge University Press:  17 September 2013

H.-C Kao  and
E. Knobloch
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Abstract

The bifurcation to one-dimensional weakly subcritical periodic patterns is described by the cubic-quintic Ginzburg-Landau equation

       At = µA + Axx + i(a1|A|2Ax + a2A2Ax*) + b|A|2A - |A|4A.

These periodic patterns may in turn become unstable through one of two different mechanisms, an Eckhaus instability or an oscillatory instability. We study the dynamics near the instability threshold in each of these cases using the corresponding modulation equations and compare the results with those obtained from direct numerical simulation of the equation. We also study the stability properties and dynamical evolution of different types of fronts present in the protosnaking region of this equation. The results provide new predictions for the dynamical properties of generic systems in the weakly subcritical regime.

Keywords

bifurcationcubic-quintic Ginzburg-Landau equationweakly subcritical periodic pattern
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 131 - 154
Copyright
© EDP Sciences, 2013

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