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 - B\'enard convection. 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 \pi/q, q\geq 4. We solve an unusual small divisor problem, and prove the existence of solutions for small parameter values.