Existence of quasipatterns solutions of the Swift-Hohenberg equation

Abstract : We consider the steady Swift-Hohenberg partial differential equation. It is a one-parameter family of PDE on the plane, modeling for example Rayleigh-Bénard con-vection. 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 π/q, q ≥ 4. We solve an unusual small divisor problem, and prove the existence of solutions for small parameter values. Notice a gap in the proof between Lemma 14 and Lemma 15. A forthcomming paper solves the problem.
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Boele Braaksma, Gérard Iooss, Laurent Stolovitch. Existence of quasipatterns solutions of the Swift-Hohenberg equation. Archive for Rational Mechanics and Analysis, Springer Verlag, 2014, 211 (3), ⟨10.1007/s00205-013-0627-7⟩. ⟨hal-01265181⟩

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