J Nonlinear Sci 20 (2010) 361-394.
On the Existence of Quasipattern Solutions of the Swift-Hohenberg Equation
(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
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.