Trans. Amer. Math. Soc. 351 (1999) 1575-1603.
Ian Melbourne
Abstract
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 Ginzburg-Landau 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>2.