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

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 PDE up to an exponentially small error.
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Gérard Iooss, Alastair Rucklidge. On the Existence of Quasipattern Solutions of the Swift–Hohenberg Equation. Journal of Nonlinear Science, Springer Verlag, 2010, ⟨10.1007/s00332-010-9063-0⟩. ⟨hal-01265174⟩

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