Abstract
In many mathematical models for pattern formation, a regular hexagonal pattern is stable in an infinite region. However, laboratory and numerical experiments are carried out in finite domains, and this imposes constraints on the possible patterns. In finite rectangular domains, it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational. In practice, it is found experimentally that in a rectangular region, patterns of distorted hexagons are often observed. This work analyses the geometry and dynamics of distorted hexagonal patterns. These patterns occur in two different types, either with a reflection symmetry, involving two wave numbers, or without symmetry, involving three different wave numbers. The relevant amplitude equations are studied to determine the detailed bifurcation structure in each case. It is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls. The transcritical bifurcation with D 3 symmetry that occurs for regular hexagons unfolds into either two pitchfork bifurcations or two saddlenode bifurcations. Numerical simulations of a model partial differential equation are also presented to illustrate the behaviour.
Highlights
Pattern formation is a topic of intensive current research; a recent volume of this journal has been dedicated to the subject [14]
It is appreciated that the amplitude equations describing the evolution and dynamics of patterns are generic, so that many different physical systems are governed by the same equations
This means that hexagons can occur at a transcritical bifurcation but rolls can only appear at a pitchfork bifurcation
Summary
Pattern formation is a topic of intensive current research; a recent volume of this journal has been dedicated to the subject [14]. Laboratory experiments on surface-tension-driven convection in square container yield a variety of puzzling patterns [9] Such irregular or non-equilateral hexagons can occur in a region of infinite horizontal extent with anisotropy, since breaking rotational invariance breaks the symmetry of the hexagons. The same amplitude equations have been studied in the context of convection in the presence of a mean flow [7,1], and for an anisotropic solidification problem [8] None of these works has provided complete bifurcation diagrams showing how the well-known picture for competition between rolls and regular hexagons is modified by the anisotropy.
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