Penta-Hepta Defect Motion in Hexagonal Patterns

Abstract

Structure and dynamics of penta-hepta defects in hexagonal patterns is studied in the framework of coupled amplitude equations for underlying plane waves. Analytical solution for phase eld of moving PHD is found in the far eld, which generalizes the static solution due to Pismen and Nepomnyashchy (1993). The mobility tensor of PHD is calculated using combined analytical and numerical approach. The results for the velocity of PHD climbing in slightly non-optimal hexagonal patterns are compared with numerical simulations of amplitude equations. Interaction of penta-hepta defects in optimal hexagonal patterns is considered.

Mean flow in hexagonal convection: stability and nonlinear dynamics

“…In the absence of mean flow, each independent PHD is found to move at a constant velocity, which vanishes at q = 0 [32,33]. In the presence of mean flow, we also find that isolated defects move at a constant velocity.…”

“…The defect velocity also depends on the wavevectors q i of the three modes making up the hexagon pattern [32]. More specifically, within equations (4,5) it depends only on the projections q i ·n i .…”

“…Similar to the nonlinear phase equations derived in [26], eqs. (31,32) are non-local. Here, however, the reason for this is the nonlocal effect of the mean flow: for β = 0 the equations become local.…”

“…We also apply a circular ramp at R = 0.4L, beyond which the phase is set to constant [32]. The interaction between pairs of PHD is characterized by the number…”

“…The instabilities typically lead to the formation of penta-hepta defects [30,31]. Their dynamics and interaction has also been studied in the absence of mean flows [32,33]. In the strongly nonlinear regime long-wave perturbations are still captured by phase equations [28].…”