Reducible actions of D4 ⋊ T2: Superlattice patterns and hidden symmetries

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 k_c. We identify branches of solutions that are periodic on a square lattice that inherits a reducible action of the symmetry group D₄ ⋊ 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.