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Steady-state bifurcation with Euclidean symmetry

Melbourne, Ian (1999) Steady-state bifurcation with Euclidean symmetry Transactions of the American Mathematical Society, 4. pp. 1575-1603.


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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 equal to or greater than 3.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Date : 1 April 1999
Additional Information : First published in Transactions of the American Mathematical Society, 4, 1575-1603, published by the American Mathematical Society. © 1999 American Mathematical Society.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:41
Last Modified : 23 Sep 2013 18:33

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