Steady-state bifurcation with Euclidean symmetry

Abstract

We consider systems of partial differential equations equivariant under the Euclidean group E ( n ) \mathbf {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 n=1 and n = 2 n=2 and for reaction-diffusion equations with general n 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 ) \mathbf {E}(n) . The representation theory of E ( n ) \mathbf {E}(n) is driven by the irreducible representations of O ( n − 1 ) \mathbf {O}(n-1) . For n = 1 n=1 , this constitutes a mathematical statement of the ‘universality’ of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When n = 2 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 ) \mathbf {O}(1) . There are infinitely many possibilities for each n ≥ 3 n\ge 3 .