Stabilization by Slow Diffusion in a Real Ginzburg-Landau System |
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Authors: | Email author" target="_blank">A?DoelmanEmail author G?Hek N?Valkhoff |
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Institution: | (1) Korteweg–deVries Instituut, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands;(2) Centrum voor Wiskunde en Informatica, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands |
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Abstract: | The Ginzburg-Landau equation is essential for understanding the
dynamics of patterns in a wide variety of physical contexts. It
governs the evolution of small amplitude instabilities near
criticality. It is well known that the (cubic) Ginzburg-Landau
equation has various unstable solitary pulse solutions. However, such
localized patterns have been observed in systems in which there are
two competing instability mechanisms. In such systems, the evolution
of instabilities is described by a Ginzburg-Landau equation coupled to
a diffusion equation.
In this article we study the influence of this additional diffusion
equation on the pulse solutions of the Ginzburg-Landau equation in
light of recently developed insights into the effects of slow diffusion
on the stability of pulses. Therefore, we consider the limit case of
slow diffusion, i.e., the situation in which the additional diffusion
equation acts on a long spatial scale. We show that the solitary
pulse solution of the Ginzburg-Landau equation persists under this
coupling. We use the Evans function method to analyze the effect of
the slow diffusion and to show that it acts as a control mechanism
that influences the (in)stability of the pulse. We establish that
this control mechanism can indeed stabilize a pulse when higher order
nonlinearities are taken into account. |
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Keywords: | |
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