%e study numerically a Swift-Hohenberg equation describing, in the weak dispersion limit, nascent optical bistability with transverse efFects. We predict that stable localized structures, and organized clusters of them, may form in the transverse plane. These structures consist of either kinks or dips. The number and spatial distribution of these localized structures are determined by the initia1 conditions while their peak (bottom) intensity remains essentia1ly constant for fixed values of the system's parameters.PACS numbers: 42.65.Pc, 42.60.Mi At the onset of optical bistability, there is a critical point where the output versus input characteristics have an infinite slope. The vicinity of this critical point is characterized by critical slowing down [1]. This implies that the dynamics of the system is dominated by a characteristic decay time which is of geometrical origin. It is inversely proportional to the deviation from the critical point and diverges at criticality. Thus in the vicinity of the critical point, all atomic and cavity decay times are associated with fast decays. Let (X, Y, C) be the deviations of the cavity field, of the injected field, and of the cooperativity parameter with respect to the values of these quantities at the critical point:Swift-Hohenberg equation [3], though some of them have been reported for other (nonvariational) models studied in chemistry and hydrodynamics [4,5], and for the cornplex Ginzburg-Landau equation [6 -9]; see [10], for a review on this topic.The situation which interests us requires that 6 ) 0 (or equivalently that a, )~, ) a,). Notably, in that case, the transverse Laplacian term in (2) is destabilizing and allows for the formation of stationary, spatially periodic patterns characterized by an intrinsic wavelength solely determined by dynamical parameters and not by the system's physical dimensions or geometrical constraints (Turing instability, [ll]). Using the t) expansion, based on the distance from the Turing bifurcation point as the smallness parameter [12], we have analytically determined the variety and the stability properties of the patterns which are solutions of Eq. (2) in the weakly nonlinear regime where the Turing bifurcation is supercritical [13]. This analysis restricted the values of the cooperativity parameter to the range X, = v3(1+id), Y, = 3V3(1+ 6'), C, = 4(1+ 6').(1)In these expressions, 6-: (u, -u, )/p~= -8 = -(~, -a, )/r, where~(~"v, ) is the atomic (external, cavity) frequency while p~a nd K are the atomic polarization and cavity decay rates. It has been demonstrated recently [2] that in the double limit of weak dispersion ([ 6 [(( 1) and nascent bistability ([ C [(( 1), the spatiotemporal evolution of the electric field X obeys an equation of the Swift-Hohenberg type: in which, furthermore, the homogeneous steady states necessarily are monostable, whatever the value of the input field y is, We proved that under those conditions, the only stable patterns forming in bidimensional transverse systems are those which either have th...

We show that spatial self-organization allows vegetation to survive greater resource limitation. Isolated vegetation patches observed in nutrient-poor territories of South America and West Africa are interpreted as localized structures arising from the bistability between the bare state and the patchy vegetation state.

Analytic and numerical investigations of a cavity containing a Kerr medium are reported. The mean field equation with plane-wave excitation and diffraction is assumed. Stable hexagons are dominant close to threshold for a self-focusing medium. Bistable switching frustrates pattern formation for a self-defocusing medium. Under appropriate parametric conditions that we identify, there is coexistence of a homogeneous stationary solution, of a hexagonal pattern solution and of a large (in principle infinite) number of localized structure solutions which connect the homogeneous and hexagonal state. Further above threshold, the hexagons show defects, and then break up with apparent turbulence. For Gaussian beam excitation, the different symmetry leads to polygon formation for narrow beams, but quasihexagonal structures appear for broader beams.

The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 1-4). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.

ABSTRACT:The large-scale vegetation patterns observed in many arid regions are due to the existence of facilitative and competitive interactions that affect the communal development of plants. For the patterns to form, it is necessary that competitive interactions be of longer range than facilitative interactions. Aridity affects the pattern's symmetry properties. As it increases, one first finds patterns constituted of spots of sparser vegetation, which then transform into an alternation of stripes of sparser and thicker vegetation, and finally into a pattern of vegetation spots separated by bare ground. The model and nonlinear analysis presented below explains the observations.

Abstract. Regular vegetation patterns appear on aerial views of plateaux in SW Niger where densely and sparsely populated zones alternate with each other. This spatial organization of the vegetation is an endogenous phenomenon which is not limited to specific plants or soils; it is a characteristic landscape of many arid regions throughout the world. The phenomenon is interpreted as the result of a spatial range difference between two biologically distinct interactions operating at the plant population level. The proposed mechanism is independent of external heterogeneities deriving from soil geomorphology or meteorology. We present a model to simulate the genesis of vegetation stripes. In addition, the model predicts the occurrence of vegetation hexagons corresponding to higher or lower density spots arranged in a hexagonal lattice. The distinction between the two spatial symmetries is discussed in terms of their Fourier transforms.

We present a model and nonlinear analysis which account for the clustering behaviors of arid vegetation ecosystems, the formation of localized bare soil spots (sometimes also called fairy circles) in these systems and the attractive or repulsive interactions governing their spatio-temporal evolution. Numerical solutions of the model closely agree with analytical predictions.

We study the properties of 2D cavity solitons in a coherently driven optical resonator subjected to a delayed feedback. The delay is found to induce a spontaneous motion of a single cavity soliton that is stationary and stable otherwise. This behavior occurs when the product of the delay time and the feedback strength exceeds some critical value. We derive an analytical formula for the speed of a moving soliton. Numerical results are in good agreement with analytical predictions.