Existence of quasipatterns solutions of the Swift-Hohenberg equation

TL;DR

This paper proves the existence of quasipattern solutions to the steady Swift-Hohenberg equation near critical parameter values, addressing a complex small divisor problem for solutions invariant under specific rotations.

Contribution

It establishes the existence of quasipattern solutions for the Swift-Hohenberg PDE near criticality, solving an unusual small divisor problem.

Findings

Existence of quasipattern solutions near critical parameter values.

Solutions are quasiperiodic and invariant under rotations of angle  g.

Addresses and resolves a complex small divisor problem.

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.