Deformed Hexagonal Patterns in a Weakly Anisotropic System

Abstract

For a t~vo-dimensional model of pattern formation the interplay between a broken up down symmetry and a weakly broken rotational symmetry is investigated. Both symmetries may be broken, for instance, in chemical reactions with an applied electric field or in thermal convection of planarly aligned nematic liquid crystals. In a system with rotational symmetry and a broken up down symmetry hexagonal patterns are favored in a certain parameter range. With increasing values for the anisotropies, by keeping the up down symmetry broken, hexagons may be deformed to centered rectangular patterns. Hence, breaking both symmetries is an alternative mechanism leading to rectangular patterns. Finally, at larger values of the anisotropies a bifurcation to stripes takes place. 1.

Two-Mode Rhomboidal States in Driven Surface Waves

“…Such states have been observed numerically in a Swift-Hohenberg model possessing two unstable wave numbers [10]. They have also been theoretically anticipated in anisotropic models where two unstable degenerate wave vectors are resonant with an externally imposed wave number [11], in nonlinear optical systems [12,13], and in the analysis of the Faraday instability excited with two frequencies [14].…”

Stripe-hexagon competition in forced pattern-forming systems with broken up-down symmetry

“…In systems with a strong axial anisotropy, such as in nematic liquid crystals, this rotational symmetry is broken [41,42,43,44] and hexagonal convection patterns do not occur. Thus the interplay between different broken symmetries, such as the broken up-down symmetry with a weak anisotropy, leads to an interesting competition between stripe and hexagonal patterns as has been shown recently [45,46,47].…”

“…which are quite similar to those presented in Refs. [35,45,54]. Depending on which of the resonant cases mentioned above is considered either G 1 or G 2 is nonzero.…”