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The effect of impurities on striped phases

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  20 August 2018

Gabriela Jaramillo ,
Arnd Scheel  and
Qiliang Wu
Gabriela Jaramillo
Affiliation:
Department of Mathematics, The University of Arizona, 617 N. Santa Rita Avenue, Tucson, AZ 85721, USA
Arnd Scheel
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN 55455, USA
Qiliang Wu
Affiliation:
Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA
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Abstract

We study the effect of algebraically localized impurities on striped phases in one spatial dimension. We therefore develop a functional-analytic framework that allows us to cast the perturbation problem as a regular Fredholm problem despite the presence of the essential spectrum, caused by the soft translational mode. Our results establish the selection of jumps in wavenumber and phase, depending on the location of the impurity and the average wavenumber in the system. We also show that, for select locations, the jump in the wavenumber vanishes.

Keywords

Turing patternsinhomogeneitiesFredholm operatoressential spectrum

MSC classification

Primary: 35K55: Nonlinear parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 1 , February 2019 , pp. 131 - 168
Copyright
Copyright © Royal Society of Edinburgh 2018 

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Footnotes

*

Present address: Department of Mathematics, Ohio University Morton Hall 321, 1 Ohio University, Athens, OH 45701, USA ([email protected]).