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Stability, bifurcations and edge oscillations in standing pulse solutions to an inhomogeneous reaction-diffusion system

Published online by Cambridge University Press:  14 November 2011

J. E. Rubin
J. E. Rubin
Affiliation:
Division of Applied MathematicsBrown UniversityBox F, Providence, RI 02912, USA
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Abstract

We consider a class of inhomogeneous systems of reaction-diffusion equations that includes a model for cavity dynamics in the semiconductor Fabry–Pérot interferometer. By adapting topological and geometrical methods, we prove that a standing pulse solution to this system is stable in a certain parameter regime, under the simplification of homogeneous illumination. Moreover, we explain two bifurcation mechanisms which can cause a loss of stability, yielding travelling and standing pulses, respectively. We compute conditions for these bifurcations to persist when inhomogeneity is restored through a certain general perturbation. Under certain of these conditions, a Hopf bifurcation results, producing periodic solutions called edge oscillations. These inhomogeneous bifurcation mechanisms represent new means for the generation of solutions displaying edge oscillations in a reaction-diffusion system. The oscillations produced by each inhomogeneous bifurcation are expected to depend qualitatively on the properties of the corresponding homogeneous bifurcation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 5 , 1999 , pp. 1033 - 1079
Copyright
Copyright © Royal Society of Edinburgh 1999

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