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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
Article

Abstract

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.

Keywords

Unit Circle Invariant Subspace Implicit Function Theorem Fourier Expansion Selfadjoint Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Argentina M., Iooss G.: Quasipatterns in a parametrically forced horizontal fluid film.. Physica D: Nonlinear Phenomena 241(16), 1306–1321 (2012)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, V.I.: Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir, 1980Google Scholar
  3. 3.
    Berti M., Bolle P., Procesi M.: An abstract Nash–Moser theorem with parameters and applications to PDEs. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(1), 377–399 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Berti, M.: Nonlinear oscillations of Hamiltonian PDEs. Progress in Nonlinear Differential Equations and their Applications, Vol. 74. Birkhäuser Boston Inc., Boston, 2007Google Scholar
  5. 5.
    Bourgain J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5(4), 629–639 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Christiansen B., Alstrom P., Levinsen M.T.: Ordered capillary-wave states: quasi-crystals, hexagons, and radial waves. Phys. Rev. Lett., 68, 2157–2160 (1992)ADSCrossRefGoogle Scholar
  7. 7.
    Craig, W.: Problèmes de petits diviseurs dans les équations aux dérivées partielles. Panoramas et Synthèses [Panoramas and Syntheses], Vol. 9. Société Mathématique de France, Paris, 2000Google Scholar
  8. 8.
    Dieudonné, J.: Foundations of Modern Analysis. Pure and Applied Mathematics, Vol. X. Academic Press, New York, 1960Google Scholar
  9. 9.
    Edwards W.S., Fauve S.: Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148 (1994)ADSCrossRefMathSciNetGoogle Scholar
  10. 10.
    Iooss G., Rucklidge A.M.: On the existence of quasipattern solutions of the Swift–Hohenberg equation. J. Nonlinear Sci. 20(3), 361–394 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin, 1995 (reprint of the 1980 edition)Google Scholar
  12. 12.
    Rucklidge, A.M., Rucklidge, W.J.: Convergence properties of the 8, 10 and 12 mode representations of quasipatterns. Physica D, 178(1–2), 62–82 (2003)Google Scholar
  13. 13.
    Rucklidge, A.M., Silber, M.: Quasipatterns in parametrically forced systems. Phys. Rev. E (3) 75(5), 055203, 4 (2007)Google Scholar

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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