Europhys. Lett, 47 (3), pp. 331-337 (1999)
On the invasion of an unstable structureless state by a stable hexagonal pattern
Z. Csahók 1,2 and C. Misbah 1
1 Laboratoire de Spectrométrie Physique, Université Joseph
Grenoble I - B.P. 87, Saint-Martin d'Hères, 38402 Cedex, France
2 MTA Rescarch Institute for Technical Physics and Materials Science
P.O.Box 49, H-1525 Budapest, Hungary
(received 17 December 1998; accepted in final form 31 May 1999)
PACS. 47.54 - Pattern selection; pattern formation.
PACS. 47.20Hw - Morphological instability; phase changes.
PACS. 47.20Ky - Nonlinearity (including bifurcation theory).
We investigate the invasion of a hexagonal pattern at the expense of an initially structureless state. We show that even in the vicinity of threshold higher-order contributions in the amplitude equations play a decisive role: the invasion velocity of the hexagonal state as evaluated from marginal stability increases. We find that the roll belt that forms around the hexagons does not widen, contrary to previous studies (L. M. Pismer et al., Europhys. Lett., 27 (1994) 433). We confirm this result by numerical simulation and also present results on the fact that even though far from the threshold hexagons exhibit complex temporal oscillations their invasion velocity is still given by the linear marginal stability criterion. Furthermore we provide a heuristic argument on the existence and the width of the roll belt.
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