Defect formation in the Swift-Hohenberg equation

Effect of heterogeneous environmental conditions on labyrinthine vegetation patterns

Self-organization is a ubiquitous phenomenon in Nature due to the permanent balance between injection and dissipation of energy. The wavelength selection process is the main issue of pattern formation. Stripe, hexagon, square, and labyrinthine patterns are observed in homogeneous conditions. In systems with heterogeneous conditions, a single wavelength is not the rule. Large-scale self-organization of vegetation in arid environments can be affected by heterogeneities, such as interannual precipitation fluctuations, fire occurrences, topographic variations, grazing, soil depth distribution, and soil-moisture islands. Here, we investigate theoretically the emergence and persistence of vegetation labyrinthine patterns in ecosystems under deterministic heterogeneous conditions. Based on a simple local vegetation model with a space-varying parameter, we show evidence of perfect and imperfect labyrinthine patterns, as well as disordered vegetation self-organization. The intensity level and the correlation of the heterogeneities control the regularity of the labyrinthine self-organization. The phase diagram and the transitions of the labyrinthine morphologies are described with the aid of their global spatial features. We also investigate the local spatial structure of labyrinths. Our theoretical findings qualitatively agree with satellite images data of arid ecosystems that show labyrinthinelike textures without a single wavelength.

Topological measures of order for imperfect two-dimensional Bravais lattices

Motivated by patterns with defects in natural and laboratory systems, we develop two quantitative measures of order for imperfect Bravais lattices in the plane. A tool from topological data analysis called persistent homology combined with the sliced Wasserstein distance, a metric on point distributions, are the key components for defining these measures. The measures generalize previous measures of order using persistent homology that were applicable only to imperfect hexagonal lattices in two dimensions. We illustrate the sensitivities of these measures to the degree of perturbation of perfect hexagonal, square, and rhombic Bravais lattices. We also study imperfect hexagonal, square, and rhombic lattices produced by numerical simulations of pattern-forming partial differential equations. These numerical experiments serve to compare the measures of lattice order and reveal differences in the evolution of the patterns in various partial differential equations.

Defect formation dynamics in curved elastic surface crystals

Topological defects shape the material and transport properties of physical systems. Examples range from vortex lines in quantum superfluids, defect-mediated buckling of graphene, and grain boundaries in ferromagnets and colloidal crystals, to domain structures formed in the early universe. The Kibble-Zurek (KZ) mechanism describes the topological defect formation in continuous non-equilibrium phase transitions with a constant finite quench rate. Universal KZ scaling laws have been verified experimentally and numerically for second-order transitions in planar Euclidean geometries, but their validity for non-thermal transitions in curved and topologically nontrivial systems still poses open questions. Here, we use recent experimentally confirmed theory to investigate topological defect formation in curved elastic surface crystals formed by stress-quenching a bilayer material. For both spherical and toroidal crystals, we find that the defect densities follow KZ-type power laws. Moreover, the nucleation sequences agree with recent experimental observations for spherical colloidal crystals. Our results suggest that curved elastic bilayers provide an experimentally accessible macroscopic system to study universal properties of non-thermal phase transitions in non-planar geometries.

Pattern formation in individual-based systems with time-varying parameters

We study the patterns generated in finite-time sweeps across symmetry-breaking bifurcations in individual-based models. Similar to the well-known Kibble-Zurek scenario of defect formation, large-scale patterns are generated when model parameters are varied slowly, whereas fast sweeps produce a large number of small domains. The symmetry breaking is triggered by intrinsic noise, originating from the discrete dynamics at the microlevel. Based on a linear-noise approximation, we calculate the characteristic length scale of these patterns. We demonstrate the applicability of this approach in a simple model of opinion dynamics, a model in evolutionary game theory with a time-dependent fitness structure, and a model of cell differentiation. Our theoretical estimates are confirmed in simulations. In further numerical work, we observe a similar phenomenon when the symmetry-breaking bifurcation is triggered by population growth.

Coarsening in potential and nonpotential models of oblique stripe patterns

We study the coarsening of two-dimensional oblique stripe patterns by numerically solving potential and nonpotential anisotropic Swift-Hohenberg equations. Close to onset, all models exhibit isotropic coarsening with a single characteristic length scale growing in time as t1/2. Further from onset, the characteristic lengths along the preferred directions x and ŷ grow with different exponents, close to 1/3 and 1/2, respectively. In this regime, one-dimensional dynamical scaling relations hold. We draw an analogy between this problem and model A in a stationary, modulated external field. For deep quenches, nonpotential effects produce a complicated dislocation dynamics that can lead to either arrested or faster-than-power-law growth, depending on the model considered. In the arrested case, small isolated domains shrink down to a finite size and fail to disappear. A comparison with available experimental results for electroconvection in nematics is presented.

Testing the Kibble-Zurek mechanism in Rayleigh-Bénard convection

We report experimental evidence of the fact that, in an emerging Rayleigh-Bénard structure, the density of defects which appear scales as a power law in the rate of change of the control parameter. The scaling exponents agree with those calculated from the Kibble-Zurek mechanism. This is the first evidence to our knowledge that this mechanism works in a hydrodynamical system.

Numerical simulations of two-dimensional magnetic domain patterns

I show that a model for the interaction of magnetic domains that includes a short range ferromagnetic and a long range dipolar antiferromagnetic interaction reproduces very well many characteristic features of two-dimensional magnetic domain patterns. In particular bubble and stripe phases are obtained, along with polygonal and labyrinthine morphologies. In addition, two puzzling phenomena, namely, the so called "memory effect" and the "topological melting" observed experimentally, are also qualitatively described. Very similar phenomenology is found in the case in which the model is changed to be represented by the Swift-Hohenberg equation driven by an external orienting field.

Hierarchical structure description of spatiotemporal chaos

We develop a hierarchical structure (HS) analysis for quantitative description of statistical states of spatially extended systems. Examples discussed here include an experimental reaction-diffusion system with Belousov-Zhabotinsky kinetics, the two-dimensional complex Ginzburg-Landau equation, and the modified FitzHugh-Nagumon equation, which all show complex dynamics of spirals and defects. We demonstrate that the spatial-temporal fluctuation fields in the above-mentioned systems all display the HS similarity property originally proposed for the study of fully developed turbulence [Phys. Rev. Lett. 72, 336 (1994)]]. The derived values of a HS parameter beta from experimental and numerical data in various physical regimes exhibit consistent trends and characterize the degree of turbulence in the systems near the transition, and the degree of heterogeneity of multiple disorders far from the transition. It is suggested that the HS analysis offers a useful quantitative description for the complex dynamics of two-dimensional spatiotemporal patterns.

Dynamics of laser-induced electroconvection pulses

We first report that, for planar nematic 4-methoxy-benzilidene-4-butylaniline (MBBA), the electroconvection threshold voltage has a nonmonotonic temperature dependence, with a well-defined minimum, and a slope of about -0.12 V/degrees C near room temperature at 70 Hz. Motivated by this observation, we have designed an experiment in which a weak continuous-wave absorbed laser beam with a diameter comparable to the pattern wavelength generates a locally supercritical region, or pulse, in dye-doped MBBA. Working 10-20 % below the laser-free threshold voltage, we observe a steady-state pulse shaped as an ellipse with the semimajor axis oriented parallel to the nematic director, with a typical size of several wavelengths. The pulse is robust, persisting even when spatially extended rolls develop in the surrounding region, and displays rolls that counterpropagate along the director at frequencies of tenths of Hz, with the rolls on the left (right) side of the ellipse moving to the right (left). Systematic measurements of the sample-voltage dependence of the pulse amplitude, spatial extent, and frequency show a saturation or decrease when the control parameter (evaluated at the center of the pulse) approaches approximately 0.3. We propose that the model for these pulses should be based on the theory of control-parameter ramps, supplemented with new terms to account for the advection of heat away from the pulse when the surrounding state becomes linearly unstable. The advection creates a negative feedback between the pulse size and the efficiency of heat transport, which we argue is responsible for the attenuation of the pulse at larger control-parameter values.