The Cross-Newell phase diffusion equation τ(k)Θ T = −∇ · kB(k), k = ∇Θ, | k| = k, and its regularization describe patterns and defects far from onset in large aspect ratio systems with translational and rotational symmetry. In this paper we show how director field solutions of this equation can be used to describe features of global patterns. The ideas are illustrated in the context of a non-trivial case study of high Prandtl number convection in a large aspect ratio, shallow, elliptical container with heated sidewalls, for which we also have the results of simulation and experiment.
Spontaneous pattern formation has been observed experimentally when two counterpropagating laser beams interact in sodium vapor. We report investigations into the nature, stability, and range of patterns in such systems. The model is an anti-reflected slab of Kerr medium, irradiated from each side by smooth, constant input fields-plane waves or Gaussian beams. Two transverse-dimensional numerical simulations have produced a rich variety of patterns. On the self-focusing side, hexagons are preferred and we find that these patterns persist to about 15% below the linear threshold intensity both with and without the wavelength-scale index grating. These results are in general agreement with our analysis and with hexagon dynamics in other situations such as fluid convection. A new phenomenon not present in typical hexagon structures is the occurrence of a Hopf bifurcation, through which the hexagon pattern destabilizes. On the self-defocusing side, squares are the favored pattern. Simulations including the index grating indicate that above threshold they are stable while below threshold they are unstable and are replaced with the homogeneous plane-wave state.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.