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Journal of Nonlinear Science

, Volume 20, Issue 3, pp 361–394 | Cite as

On the Existence of Quasipattern Solutions of the Swift–Hohenberg Equation

  • G. Iooss
  • A. M. Rucklidge
Article

Abstract

Quasipatterns (two-dimensional patterns that are quasiperiodic in any spatial direction) remain one of the outstanding problems of pattern formation. As with problems involving quasiperiodicity, there is a small divisor problem. In this paper, we consider 8-fold, 10-fold, 12-fold, and higher order quasipattern solutions of the Swift–Hohenberg equation. We prove that a formal solution, given by a divergent series, may be used to build a smooth quasiperiodic function which is an approximate solution of the pattern-forming partial differential equation (PDE) up to an exponentially small error.

Keywords

Bifurcations Quasipattern Small divisors Gevrey series 

Mathematics Subject Classification (2000)

35B32 35C20 40G10 52C23 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.I.U.F.Université de NiceNiceFrance
  2. 2.Department of Applied MathematicsUniversity of LeedsLeedsEngland

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