Defects in roll-hexagon competition

Abstract

The defects of a system where hexagons and rolls are both stable solutions are considered. On the basis of topological arguments we show that the unstable phase is present in the core of the defects. This means that a roll is present in the penta-hepta defect of hexagons and that a hexagon is found in the core of a grain boundary connecting rolls with different orientations. These results are verified in an experiment of thermal convection under non-Boussinesq conditions. PACS numbers: 47.20. Bp, 47.25.Qv Defects play an important role in the dynamics of pattern-forming systems. Specifically, dislocations and grain boundaries in convective patterns of rolls, and spirals and centered defects in chemical reactions, have been the object of several studies. ! However, the structure of defects has not been carefully analyzed in systems where two different symmetries coexist. This is a very important case that appears very often in nature, a typical example being the transition between hexagons and rolls in thermal convection. The competition between patterns associated with different symmetries has recently been discussed on the basis of general arguments. 2 The purpose of this Letter is to study defect properties when hexagons and rolls are stable solutions in a nonequilibrium pattern-forming system.The competition between hexagons and rolls can be described by means of three coupled Ginzburg-Landau equations (GLH), which determine the behavior of the three complex amplitudes A t of the sets of rolls describing the hexagonal structure. Each of them makes an angle of 2;r/3 with each of the others. A qualitative description of the nature of the cores of the various defects which may be observed in this problem can be deduced 3 from an elementary study of the following sixdimensional dynamical system, obtained from GLH, in the limit of homogeneous patterns: 4 d

Transition to chemical turbulence

“…10). in an analysis of amplitUde equations, Ciliberto et al 42 showed that a grain boundary between two oblique rolls (stripes) gives rise locally to hexagons, and that, in the defect of hexagons, a roll (stripe) is present. We conjecture that in a parameter region where stripes and hexagons are nearly equally sta-.…”

Geometric diagnostics of complex patterns: Spiral defect chaos

“…20 Many complex patterns exhibit striking, isolated objects like point defects, e.g., dislocations, 21,22 disclinations, 23 or penta-hepta defects. 24,25 Dislocations often take on the form of spirals, as is the case for points of vanishing oscillation amplitude in spatially extended oscillations. 26,27 Similar spirals also arise in excitable systems.…”

Penta-Hepta Defect Motion in Hexagonal Patterns

“…Equations of motion (4) indicate that the penta-hepta defect is driven by superposition of two Peach-K ohler forces corresponding to a strain at singular modes 2,3. If K 2;3 = 0, defect does not move.…”