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On the Existence of Quasipattern Solutions of the Swift-Hohenberg Equation

Iooss, G and Rucklidge, AM (2010) On the Existence of Quasipattern Solutions of the Swift-Hohenberg Equation. Journal of Nonlinear Science, 20 (3). 361 - 394 . ISSN 0938-8974


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Quasipatterns (two-dimensional patterns that are quasiperiodic in any spatial direction) remain one of the outstanding problems of pattern formation. As with problems involving quasiperiodicity, there is a small divisor problem. In this paper, we consider 8-fold, 10-fold, 12-fold, and higher order quasipattern solutions of the Swift-Hohenberg equation. We prove that a formal solution, given by a divergent series, may be used to build a smooth quasiperiodic function which is an approximate solution of the pattern-forming partial differential equation (PDE) up to an exponentially small error.

Item Type: Article
Copyright, Publisher and Additional Information: © 2010 Springer. Reproduced in accordance with the publisher's self-archiving policy.
Keywords: Bifurcations, Quasipattern, Small divisors, Gevrey series, Quasi-periodic patterns, Faraday waves, Crystals, Gravity
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 12 Oct 2011 11:53
Last Modified: 08 Feb 2013 17:34
Published Version: http://dx.doi.org/10.1007/s00332-010-9063-0
Status: Published
Publisher: Springer
Refereed: Yes
Identification Number: 10.1007/s00332-010-9063-0
URI: http://eprints.whiterose.ac.uk/id/eprint/43324

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