Steady-state Bifurcation with Euclidean Symmetry

We consider systems of partial differential equations equivariant under the Euclidean group E(n) and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when n = 1 and n = 2 and for reaction-diffusion equations with general n, reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.) In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of E(n). The representation theory of E(n) is driven by the irreducible representations of O(n − 1). For n = 1, this constitutes a mathematical statement of the ‘universality’ of the GinzburgLandau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When n = 2, there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of O(1). There are infinitely many possibilities for each n ≥ 3.