Rotating Convection in an Anisotropic System

We study the stability of patterns arising in rotating convection in weakly anisotropic systems using a modified Swift-Hohenberg equation. The anisotropy, either an endogenous characteristic of the system or induced by external forcing, can stabilize periodic rolls in the K\"uppers-Lortz chaotic regime. For the particular case of rotating convection with time-modulated rotation where recently, in experiment, chiral patterns have been observed in otherwise K\"uppers-Lortz-unstable regimes, we show how the underlying base-flow breaks the isotropy, thereby affecting the linear growth-rate of convection rolls in such a way as to stabilize spirals and targets. Throughout we compare analytical results to numerical simulations of the Swift-Hohenberg equation

Spatially Extended Dislocations Produced by the Dispersive Swift-Hohenberg Equation

Motivated by previous results showing that the addition of a linear dispersive term to the two-dimensional Kuramoto-Sivashinsky equation has a dramatic effect on the pattern formation, we study the Swift-Hohenberg equation with an added linear dispersive term, the dispersive Swift-Hohenberg equation (DSHE). The DSHE produces stripe patterns with spatially extended dislocations that we call seam defects. In contrast to the dispersive Kuramoto-Sivashinsky equation, the DSHE has a narrow band of unstable wavelengths close to an instability threshold. This allows for analytical progress to be made. We show that the amplitude equation for the DSHE close to threshold is a special case of the anisotropic complex Ginzburg-Landau equation (ACGLE) and that seams in the DSHE correspond to spiral waves in the ACGLE. Seam defects and the corresponding spiral waves tend to organize themselves into chains, and we obtain formulas for the velocity of the spiral wave cores and for the spacing between them. In the limit of strong dispersion, a perturbative analysis yields a relationship between the amplitude and wavelength of a stripe pattern and its propagation velocity. Numerical integrations of the ACGLE and the DSHE confirm these analytical results

Grain boundary motion in layered phases

We study the motion of a grain boundary that separates two sets of mutually perpendicular rolls in Rayleigh-B\'enard convection above onset. The problem is treated either analytically from the corresponding amplitude equations, or numerically by solving the Swift-Hohenberg equation. We find that if the rolls are curved by a slow transversal modulation, a net translation of the boundary follows. We show analytically that although this motion is a nonlinear effect, it occurs in a time scale much shorter than that of the linear relaxation of the curved rolls. The total distance traveled by the boundary scales as $\epsilon^{-1/2}$, where $\epsilon$ is the reduced Rayleigh number. We obtain analytical expressions for the relaxation rate of the modulation and for the time dependent traveling velocity of the boundary, and especially their dependence on wavenumber. The results agree well with direct numerical solutions of the Swift-Hohenberg equation. We finally discuss the implications of our results on the coarsening rate of an ensemble of differently oriented domains in which grain boundary motion through curved rolls is the dominant coarsening mechanism.Comment: 16 pages, 5 figure

Swift-Hohenberg model for magnetoconvection

A model system of partial differential equations in two dimensions is derived from the three-dimensional equations for thermal convection in a horizontal fluid layer in a vertical magnetic field. The model consists of an equation of Swift-Hohenberg type for the amplitude of convection, coupled to an equation for a large-scale mode representing the local strength of the magnetic field. The model facilitates both analytical and numerical studies of magnetoconvection in large domains. In particular, we investigate the phenomenon of flux separation, where the domain divides into regions of strong convection with a weak magnetic field and regions of weak convection with a strong field. Analytical predictions of flux separation based on weakly nonlinear analysis are extended into the fully nonlinear regime through numerical simulations. The results of the model are compared with simulations of the full three-dimensional magnetoconvection problem.S. M. Cox, P. C. Matthews, and S. L. Pollicot

Snakes and ladders in an inhomogeneous neural field model

Continuous neural field models with inhomogeneous synaptic connectivities are known to support traveling fronts as well as stable bumps of localized activity. We analyze stationary localized structures in a neural field model with periodic modulation of the synaptic connectivity kernel and find that they are arranged in a snakes-and-ladders bifurcation structure. In the case of Heaviside firing rates, we construct analytically symmetric and asymmetric states and hence derive closed-form expressions for the corresponding bifurcation diagrams. We show that the ideas proposed by Beck and co-workers to analyze snaking solutions to the Swift-Hohenberg equation remain valid for the neural field model, even though the corresponding spatial-dynamical formulation is non-autonomous. We investigate how the modulation amplitude affects the bifurcation structure and compare numerical calculations for steep sigmoidal firing rates with analytic predictions valid in the Heaviside limit

Pattern forming pulled fronts: bounds and universal convergence

We analyze the dynamics of pattern forming fronts which propagate into an unstable state, and whose dynamics is of the pulled type, so that their asymptotic speed is equal to the linear spreading speed v^*. We discuss a method that allows to derive bounds on the front velocity, and which hence can be used to prove for, among others, the Swift-Hohenberg equation, the Extended Fisher-Kolmogorov equation and the cubic Complex Ginzburg-Landau equation, that the dynamically relevant fronts are of the pulled type. In addition, we generalize the derivation of the universal power law convergence of the dynamics of uniformly translating pulled fronts to both coherent and incoherent pattern forming fronts. The analysis is based on a matching analysis of the dynamics in the leading edge of the front, to the behavior imposed by the nonlinear region behind it. Numerical simulations of fronts in the Swift-Hohenberg equation are in full accord with our analytical predictions.Comment: 27 pages, 9 figure