Stability analysis of line patterns of an anisotropic interaction model

Motivated by the formation of fingerprint patterns, we consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In addition, the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. Central to this pattern formation are straight line patterns. For a given spatially homogeneous tensor field, we show that there exists a preferred direction of straight lines, i.e., straight vertical lines can be stable for sufficiently many particles, while many other rotations of the straight lines are unstable steady states, both for a sufficiently large number of particles and in the continuum limit. For straight vertical lines we consider specific force coefficients for the stability analysis of steady states, show that stability can be achieved for exponentially decaying force coefficients for a sufficiently large number of particles, and relate these results to the Kücken--Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.The work of the first author was partially supported by the EPSRC through grant EP/P031587/1. The work of the second author was supported by the Leverhulme Trust research project grant ``Novel discretizations for higher order nonlinear PDE"" (RPG-2015-69). The work of the third author was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German Academic Scholarship Foundation (Studienstiftung des Deutschen Volkes). The work of the fourth author was supported by the Leverhulme Trust project on breaking the non-convexity barrier, EPSRC grant EP/M00483X/1, the EPSRC Centre EP/N014588/1, the RISE projects CHiPS and NoMADS, the Cantab Capital Institute for the Mathematics of Information, and the Alan Turing Institute

Domain wall dynamics in an optical Kerr cavity

An anisotropic (dichroic) optical cavity containing a self-focusing Kerr medium is shown to display a bifurcation between static --Ising-- and moving --Bloch-- domain walls, the so-called nonequilibrium Ising-Bloch transition (NIB). Bloch walls can show regular or irregular temporal behaviour, in particular, bursting and spiking. These phenomena are interpreted in terms of the spatio-temporal dynamics of the extended patterns connected by the wall, which display complex dynamical behaviour as well. Domain wall interaction, including the formation of bound states is also addressed.Comment: 15 pages Tex file with 11 postscript figures. Resubmitted to Phys. Rev.

Symmetry considerations and development of pinwheels in visual maps

Neurons in the visual cortex respond best to rod-like stimuli of given orientation. While the preferred orientation varies continuously across most of the cortex, there are prominent pinwheel centers around which all orientations a re present. Oriented segments abound in natural images, and tend to be collinear}; neurons are also more likely to be connected if their preferred orientations are aligned to their topographic separation. These are indications of a reduced symmetry requiring joint rotations of both orientation preference and the underl ying topography. We verify that this requirement extends to cortical maps of mo nkey and cat by direct statistical analysis. Furthermore, analytical arguments and numerical studies indicate that pinwheels are generically stable in evolving field models which couple orientation and topography

Formation of adsorbate structures induced by external electric field in plasma-condensate systems

We present a new model of plasma-condensate system, by taking into account an anisotropy of transference reactions of adatoms between neighbor layers of multi-layer system, caused by the strength of the electric field near substrate. We discuss an influence of the strength of the electric field onto first-order phase transitions and conditions for adsorbate patterning in plasma-condensate systems. It is shown that separated pyramidal-like multi-layer adsorbate islands can be formed in the plasma-condensate system if the strength of the electric field near substrate becomes larger tan the critical value, which depends on the interaction energy of adsorbate and adsorption coefficient.Comment: 8 pages, 8 figure

Double step structure and meandering due to the many body interaction at GaN(0001) surface in N-rich conditions

Growth of gallium nitride on GaN(0001) surface is modeled by Monte Carlo method. Simulated growth is conducted in N-rich conditions, hence it is controlled by Ga atoms surface diffusion. It is shown that dominating four-body interactions of Ga atoms can cause step flow anisotropy. Kinetic Monte Carlo simulations show that parallel steps with periodic boundary conditions form double terrace structures, whereas initially V -shaped parallel step train initially bends and then every second step moves forward, building regular, stationary ordering as observed during MOVPE or HVPE growth of GaN layers. These two phenomena recover surface meandered pair step pattern observed, since 1953, on many semiconductor surfaces, such as SiC, Si or GaN. Change of terrace width or step orientation particle diffusion jump barriers leads either to step meandering or surface roughening. Additionally it is shown that step behavior changes with the Schwoebel barrier height. Furthermore, simulations under conditions corresponding to very high external particle flux result in triangular islands grown at the terraces. All structures, emerging in the simulations, have their corresponding cases in the experimental results.Comment: 25 pages, 8 figure

Domain Structures in Fourth-Order Phase and Ginzburg-Landau Equations

In pattern-forming systems, competition between patterns with different wave numbers can lead to domain structures, which consist of regions with differing wave numbers separated by domain walls. For domain structures well above threshold we employ the appropriate phase equation and obtain detailed qualitative agreement with recent experiments. Close to threshold a fourth-order Ginzburg-Landau equation is used which describes a steady bifurcation in systems with two competing critical wave numbers. The existence and stability regime of domain structures is found to be very intricate due to interactions with other modes. In contrast to the phase equation the Ginzburg-Landau equation allows a spatially oscillatory interaction of the domain walls. Thus, close to threshold domain structures need not undergo the coarsening dynamics found in the phase equation far above threshold, and can be stable even without phase conservation. We study their regime of stability as a function of their (quantized) length. Domain structures are related to zig-zags in two-dimensional systems. The latter are therefore expected to be stable only when quenched far enough beyond the zig-zag instability.Comment: Submitted to Physica D, 11 pages (RevTeX 3), 12 postscript figure

Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models

We describe a generic theoretical framework, denoted as the Boltzmann-Ginzburg-Landau approach, to derive continuous equations for the polar and/or nematic order parameters describing the large scale behavior of assemblies of point-like active particles interacting through polar or nematic alignment rules. Our study encompasses three main classes of dry active systems, namely polar particles with 'ferromagnetic' alignment (like the original Vicsek model), nematic particles with nematic alignment ("active nematics"), and polar particles with nematic alignment ("self-propelled rods"). The Boltzmann-Ginzburg-Landau approach combines a low-density description in the form of a Boltzmann equation, with a Ginzburg-Landau-type expansion close to the instability threshold of the disordered state. We provide the generic form of the continuous equations obtained for each class, and comment on the relationships and differences with other approaches.Comment: 30 pages, 3 figures, to appear in Eur. Phys. J. Special Topics, in a Discussion and Debate issue on active matte