Steady-state bifurcation with Euclidean symmetry

Author:
Ian Melbourne

Journal:
Trans. Amer. Math. Soc. **351** (1999), 1575-1603

MSC (1991):
Primary 58F14, 35B32; Secondary 35K55, 35Q55

DOI:
https://doi.org/10.1090/S0002-9947-99-02147-9

MathSciNet review:
1467473

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider systems of partial differential equations equivariant under the Euclidean group $\mathbf {E}(n)$ 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 $n=1$ 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 $\mathbf {E}(n)$. The representation theory of $\mathbf {E}(n)$ is driven by the irreducible representations of $\mathbf {O}(n-1)$. 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 $\mathbf {O}(1)$. There are infinitely many possibilities for each $n\ge 3$.

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Additional Information

**Ian Melbourne**

Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3476

MR Author ID:
123300

Email:
ism@math.uh.edu

Received by editor(s):
May 22, 1996

Received by editor(s) in revised form:
December 6, 1996

Additional Notes:
Supported in part by NSF Grant DMS-9403624 and by ONR Grant N00014-94-1-0317

Article copyright:
© Copyright 1999
American Mathematical Society