Archive for Rational Mechanics and Analysis

, Volume 209, Issue 1, pp 255–285 | Cite as

Existence of Quasipattern Solutions of the Swift–Hohenberg Equation

  • Boele Braaksma
  • Gérard IoossEmail author
  • Laurent Stolovitch


We consider the steady Swift–Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh–Bénard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle \({\pi/q, q \geqq 4}\). We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.


Unit Circle Invariant Subspace Implicit Function Theorem Fourier Expansion Selfadjoint Operator 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Boele Braaksma
    • 1
  • Gérard Iooss
    • 2
    Email author
  • Laurent Stolovitch
    • 3
  1. 1.Johann Bernoulli Institute, University of GroningenGroningenThe Netherlands
  2. 2.Institut Universitaire de France-Laboratoire J.-A. Dieudonné U.M.R. 7351Université de Nice-Sophia AntipolisNice Cedex 02France
  3. 3.CNRS-Laboratoire J.-A. Dieudonné U.M.R. 7351Université de Nice-Sophia AntipolisNice Cedex 02France

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