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Essential instability of pulses and bifurcations to modulated travelling waves

Published online by Cambridge University Press:  14 November 2011

B. Sandstede  and
A. Scheel
B. Sandstede
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210, USA
A. Scheel
Affiliation:
Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, 14195 Berlin, Germany
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Extract

Reaction-diffusion systems on the real line are considered. Localized travelling waves become unstable when the essential spectrum of the linearization about them crosses the imaginary axis. In this article, it is shown that this transition to instability is accompanied by the bifurcation of a family of large patterns that are a superposition of the primary travelling wave with steady spatially periodic patterns of small amplitude. The bifurcating patterns can be parametrized by the wavelength of the steady patterns; they are time-periodic in a moving frame. A major difficulty in analysing this bifurcation is its genuinely infinite-dimensional nature. In particular, finite-dimensional Lyapunov–Schmidt reductions or centre-manifold theory do not seem to be applicable to pulses having their essential spectrum touching the imaginary axis.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 6 , 1999 , pp. 1263 - 1290
Copyright
Copyright © Royal Society of Edinburgh 1999

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