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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Steady-state bifurcation with Euclidean symmetry
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by Ian Melbourne PDF
Trans. Amer. Math. Soc. 351 (1999), 1575-1603 Request permission

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
  • © Copyright 1999 American Mathematical Society
  • 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