Proof of Quasipatterns for the Swift–Hohenberg Equation

Abstract

This paper establishes the existence of quasipatterns solutions of the Swift-Hohenberg PDE. In a former approach [BIS], we avoided the use of Nash-Moser scheme, but our proof contains a gap. The present proof of existence is based on the works by Berti et al [BBP10], [BB10], [BCP] related to the Nash-Moser scheme. For solving the small divisor problem, we need to introduce a new free parameter related to the freedom in the choice of parameterization of the bifurcating solution. Thanks to a transversality condition, the result gives only a bifurcating set, located in a small hornlike region centered on a curve, with the origin at the bifurcation point.

Existence of quasipatterns in the superposition of two hexagonal patterns

“…and that H s is an algebra for s > 2 (see [5]), and possesses the usual properties of Sobolev spaces H s in dimension 4. We prove in Appendix the following useful Lemmas: Lemma 5 For nearly all α ∈ (0, π/6), in particular for α ∈ E Q = Qπ∩]0, π/6], the only solutions of |k(z)| = 1 are ±k j , ±k ′ j j = 1, 2 and k = ±k 3 , or ±k ′ 3 , i.e.…”

“…This is done in section 3. In what follows, we only mention the differences with respect to the simple case treated in [6]. In addition to the two basic bifurcating hexagonal patterns which exist for all α, the result on the existence of quasipatterns is summed up at Theorem 28.…”

Spatiotemporal chaos and quasipatterns in coupled reaction–diffusion systems

“…Doing this for complex periodic patterns is challenging because dozens of modes are involved. For quasipatterns, there is the additional complication that this process, where the small-amplitude pattern is expressed as a power series in a small parameter, leads to divergent series [30,31], though existence of quasipatterns has been proved in the Swift-Hohenberg equation [32] and in Rayleigh-Bénard convection [33]. The case of a potentially infinite set of modes ( Figure 3c) is challenging.…”

An Iterative Method for Time-Fractional Swift-Hohenberg Equation

“…The S-H equation plays an important role in pattern formation theory. In [34], Braaksma et al proved the existence of quasipatterns for the S-H equation. Also, wave process described by the S-H equation is important as well.…”