Steady-state bifurcation with Euclidean symmetry
Transactions of the American Mathematical Society, 351 (4):
1575--1603 (1999)DOI: 10.1090/s0002-9947-99-02147-9
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 $n3$.