J Nonlinear Sci 20 (2010) 361-394. doi:10.1007/s00332-010-9063-0

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

G. Iooss(1) and A.M.Rucklidge(2).

(1) I.U.F., Universite de Nice,
Labo J.A. Dieudonne, Parc Valrose, 06108 Nice, France
(2) Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK

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.