Steady-state bifurcation with Euclidean symmetry

We consider systems of partial differential equations equivariant under the Euclidean group 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 and and for reaction-diffusion equations with general , 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 . The representation theory of is driven by the irreducible representations of . For , 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 , there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of . There are infinitely many possibilities for each .