16 (2003) 615-645.
Reducible actions of D4 x T2: superlattice patterns and hidden symmetries
P.C. Matthews(2) and
(1) Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge, CB3 9EW, UK
(2) School of Mathematical Sciences, University of Nottingham,
University Park, Nottingham, NG7 2RD, UK
(3) Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK
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 . 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.
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