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Reducible actions of D4 x T2: superlattice patterns and hidden symmetries

Dawes, J.H.P., Matthews, P.C. and Rucklidge, A.M. (2003) Reducible actions of D4 x T2: superlattice patterns and hidden symmetries. Nonlinearity, 16 (2). 615 -645. ISSN 1361-6544

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Abstract

We study steady-state pattern-forming instabilities on R2. A uniform initial state that is invariant under the Euclidean group E(2) of translations, rotations and reflections of the plane loses linear stability to perturbations with a non-zero wavenumber kc. We identify branches of solutions that are periodic on a square lattice that inherits a reducible action of the symmetry group D4 x T2. Reducible group actions occur naturally when we consider solutions that are periodic on real-space lattices that are much more widely spaced than the wavelength of the pattern-forming instability. They thus apply directly to computations in large domains where periodic boundary conditions are applied. The normal form for the bifurcation is calculated, taking the presence of various hidden symmetries into account and making use of previous work by Crawford [8]. We compute the stability (relative to other branches of solutions that exist on this lattice) of the solution branches that we can guarantee by applying the equivariant branching lemma. These computations involve terms higher than third order in the normal form, and are affected by the hidden symmetries. The effects of hidden symmetries that we elucidate are relevant also to bifurcations from fully nonlinear patterns.

In addition, other primary branches of solutions with submaximal symmetry are found always to exist; their existence cannot be deduced by applying the equivariant branching lemma. These branches are stable in open regions of the space of normal form coefficients.

The relevance of these results is illustrated by numerical simulations of a simple pattern-forming PDE.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 2003 IOP Publishing Ltd and LMS Publishing Ltd. This is an author produced version of an article published in Nonlinearity. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.
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: A. M. Rucklidge
Date Deposited: 10 Feb 2006
Last Modified: 06 Jun 2014 06:55
Published Version: http://www.iop.org/EJ/abstract/0951-7715/16/2/315/
Status: Published
Refereed: Yes
Identification Number: 10.1088/0951-7715/16/2/315
URI: http://eprints.whiterose.ac.uk/id/eprint/995

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