On the existence of quasipattern solutions of the Swift-Hohenberg equation

TL;DR

This paper investigates the existence of quasipattern solutions in the Swift-Hohenberg equation, demonstrating that divergent series can be used to construct smooth approximate solutions with exponentially small errors, addressing a key problem in pattern formation.

Contribution

It proves that formal divergent series solutions can be transformed into smooth quasiperiodic solutions for the Swift-Hohenberg equation, advancing understanding of quasipattern existence.

Findings

Formal divergent series can generate smooth quasiperiodic solutions.

Approximate solutions are accurate up to exponentially small errors.

Addresses the small divisor problem in pattern formation.

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 PDE up to an exponentially small error.