Normal form for spatial dynamics in the Swift-Hohenberg equation

John Burke; Edgar Knobloch

doi:10.3934/proc.2007.2007.170

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2007, Volume 2007: 170-180. Doi: 10.3934/proc.2007.2007.170 open access

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Normal form for spatial dynamics in the Swift-Hohenberg equation

John Burke¹ and
Edgar Knobloch^2,

1.
University of California, Department of Physics, Berkeley, CA 94720
2.
Department of Physics, University of California, Berkeley, CA 94720

Received: July 31, 2006

Revised: December 31, 2006

Published: August 2007

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Abstract

The reversible Hopf bifurcation with 1:1 resonance holds the key to the presence of spatially localized steady states in many partial differential equations on the real line. Two different techniques for computing the normal form for this bifurcation are described and applied to the Swift-Hohenberg equation with cubic/quintic and quadratic/cubic nonlinearities.

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Mathematics Subject Classification: Primary: 37G05, 37L10; Secondary: 70K45.

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