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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 1467473
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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

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