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In hyperarid environments, vegetation is highly fragmented, with plant populations exhibiting non-random biphasic structures where regions of high biomass density are separated by bare soil. In the Atacama Desert of northern Chile, rainfall is virtually nonexistent, but fog pushed in from the interior sustains patches of vegetation in a barren environment. Tillandsia landbeckii, a plant with no functional roots, survives entirely on fog corridors as a water source. Their origin is attributed to interaction feedback among the ecosystem agents, which have different spatial scales, ultimately generating banded patterns as a self-organising response to resource scarcity. The interaction feedback between the plants can be nonreciprocal due to the fact that the fog flows in a well-defined direction. Using remote sensing analysis and mathematical modelling, we characterise the orientation angle of banded vegetation patterns with respect to fog direction and topographic slope gradient. We show that banded vegetation patterns can be either oblique or horizontal to the fog flow rather than topography. The initial and boundary conditions determine the type of the pattern. The bifurcation diagram for both patterns is established. The theoretical predictions are in agreement with observations from remote sensing image analysis.

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We consider arrays of coupled nonlinear optical cavities subject to coherent optical injection. These devices are described by the discrete generalized Lugiato-Lefever equation. We predict that stable three-dimensional localized structures, often called discrete light bullets, and clusters of them may form in the output of the coupled optical resonators. We consider both anomalous and normal dispersion and show that it results in the generation of, respectively, bright and dark discrete light bullets.

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This Focus Issue on instabilities and nonequilibrium structures includes invited contributions from leading researchers across many different fields. The issue was inspired in part by the "VII Instabilities and Nonequilibrium Structures 2019" conference that took place at the Pontifica Universidad Católica de Valparaiso, Chile in December 2019. The conference, which is devoted to nonlinear science, is one of the oldest conferences in South America (since December 1985). This session has an exceptional character since it coincides with the 80th anniversary of Professor Enrique Tirapegui. We take this opportunity to highlight Tirapegui's groundbreaking contributions in the field of random perturbations experienced by macroscopic systems and in the formation of spatiotemporal structures in such systems operating far from thermodynamic equilibrium. This issue addresses a cross-disciplinary area of research as can be witnessed by the diversity of systems considered from inert matter such as photonics, chemistry, and fluid dynamics, to biology.

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We investigate and review the formation of two-dimensional dissipative rogue waves in cavity nonlinear optics with transverse effects. Two spatially extended systems are considered for this purpose: the driven Kerr optical cavities subjected to optical injection and the broad-area surface-emitting lasers with a saturable absorber. We also consider a quasi-two-dimensional system (the two dimensions being space and time) of a fiber laser describing the complex cubic-quintic Ginzburg-Landau equation. We show that rogue waves are controllable by means of time-delayed feedback and optical injection. We show that without delayed feedback, transverse structures are stationary or oscillating. However, when the strength of the delayed feedback is increased, all the systems generate giant two-dimensional pulses that appear with low probability and suddenly appear and disappear. We characterize their formation by computing the probability distribution, which shows a long tail. Besides, we have computed the significant wave height, which measures the mean wave height of the highest third of the waves. We show that for all systems, the distribution tails expand beyond two times the significant wave height. Furthermore, we also show that optical injection may suppress the rogue wave formation in a semiconductor laser with a saturable absorber.

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We introduce a spin-flip model for a broad-area vertical-cavity surface-emitting laser (VCSEL) with a saturable absorber. We demonstrate simultaneous existence of orthogonally linearly polarized and elliptically polarized cavity solitons. We show that polarization degree of freedom leads to a period-doubling route to spatially localized chaos of the elliptically polarized cavity solitons.

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Time-delayed feedback plays an important role in the dynamics of spatially extended systems. In this contribution, we consider the generic Lugiato-Lefever model with delay feedback that describes Kerr optical frequency comb in all fiber cavities. We show that the delay feedback strongly impacts the spatiotemporal dynamical behavior resulting from modulational instability by (i) reducing the threshold associated with modulational instability and by (ii) decreasing the critical frequency at the onset of this instability. We show that for moderate input intensities it is possible to generate drifting cavity solitons with an asymmetric radiation emitted from the soliton tails. Finally, we characterize the formation of rogue waves induced by the delay feedback.

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We theoretically investigate a weakly birefringent all-fiber cavity subject to linearly polarized optical injection. We describe the propagation of light inside the cavity using, for each linear polarization component of the electric field, the Lugiato-Lefever model. These two components are coupled by cross-phase modulation. We show that, for a wide range of parameters, there is a coexistence between a homogeneous steady state and two different types of temporal vector cavity solitons, which can be hosted in the same system. They differ by their polarization state and peak intensity. We construct their bifurcation diagram and show that they are connected through a saddle-node bifurcation. Finally, we show that vector cavity solitons exhibit multistability involving different polarization states with different energies.

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We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second-order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial Gaussian pumping beam and subjected to time-delayed feedback. The Gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assuming a continuous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use an order parameter approach to derive a normal form of the delay-induced Hopf bifurcation leading to an oscillating solution. Additionally we model the interplay of an attracting inhomogeneity and destabilizing time delay by describing the localized solution as an overdamped particle in a potential well generated by the inhomogeneity. In this case, the time-delayed feedback acts as a driving force. Comparing results from the later approach with the full Swift-Hohenberg model, we show that the approach not only provides an instructive description of the depinning dynamics, but also is numerically accurate throughout most of the parameter regime.

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We demonstrate a way to generate two-dimensional rogue waves in two types of broad area nonlinear optical systems subject to time-delayed feedback: in the generic Lugiato-Lefever model and in the model of a broad-area surface-emitting laser with saturable absorber. The delayed feedback is found to induce a spontaneous formation of rogue waves. In the absence of delayed feedback, spatial pulses are stationary. The rogue waves are exited and controlled by the delay feedback. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical contribution, which is often considered as a signature of the presence of rogue waves, appears for sufficiently strong feedback. The generality of our analysis suggests that the feedback induced instability leading to the spontaneous formation of two-dimensional rogue waves is a universal phenomenon.

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Desertification due to climate change and increasing drought periods is a worldwide problem for both ecology and economy. Our ability to understand how vegetation manages to survive and propagate through arid and semiarid ecosystems may be useful in the development of future strategies to prevent desertification, preserve flora-and fauna within-or even make use of scarce resources soils. In this paper, we study a robust phenomena observed in semi-arid ecosystems, by which localized vegetation patches split in a process called self-replication. Localized patches of vegetation are visible in nature at various spatial scales. Even though they have been described in literature, their growth mechanisms remain largely unexplored. Here, we develop an innovative statistical analysis based on real field observations to show that patches may exhibit deformation and splitting. This growth mechanism is opposite to the desertification since it allows to repopulate territories devoid of vegetation. We investigate these aspects by characterizing quantitatively, with a simple mathematical model, a new class of instabilities that lead to the self-replication phenomenon observed.

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We report the experimental observation of two-dimensional vector cavity solitons in a Vertical-Cavity Surface-Emitting Laser (VCSEL) under linearly polarized optical injection when varying optical injection linear polarization direction. The polarization of the cavity soliton is not the one of the optical injection as it acquires a distinct ellipticity. These experimental results are qualitatively reproduced by the spin-flip VCSEL model. Our findings open the road to polarization multiplexing when using cavity solitons in broad-area lasers as pixels in information technology.

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We investigate spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subject to time-delayed optical feedback. We show that the inclusion of feedback leads to a period doubling route to chaos of spatially localized light structures.

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We report experimental evidence of spontaneous formation of localized structures in a 80 µm diameter Vertical-Cavity Surface-Emitting Laser (VCSEL) biased above the lasing threshold and under optical injection. Such localized structures are bistable with the injected beam power and the VCSEL current. We experimentally investigate the formation of localized structures for different detunings between the injected beam and the VCSEL, and different injection beam waists.

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We study the formation of localized structures, often called localized spots, in reaction-diffusion systems subject to time delayed feedback control. We focus on the regime close to a second-order critical point marking the onset of a hysteresis loop. We show that the space-time dynamics of the FitzHugh-Nagumo model in the vicinity of that critical point could be described by the delayed Swift-Hohenberg equation. We show that the delayed feedback induces a spontaneous motion of localized spots. We characterize this motion by computing analytically the velocity and the threshold above which localized structures start to move in an arbitrary direction. Numerical solutions of the governing equation are in close agreement with those obtained from the delayed Swift-Hohenberg equation.

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Using statistical thermodynamics, we derive a general expression of the stationary probability distribution for thermodynamic systems driven out of equilibrium by several thermodynamic forces. The local equilibrium is defined by imposing the minimum entropy production and the maximum entropy principle under the scale invariance restrictions. The obtained probability distribution presents a singularity that has immediate physical interpretation in terms of the intermittency models. The derived reference probability distribution function is interpreted as time and ensemble average of the real physical one. A generic family of stochastic processes describing noise-driven intermittency, where the stationary density distribution coincides exactly with the one resulted from entropy maximization, is presented.

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In a recent paper, Phys. Rev. E 81, 041137 (2010), the author attempts to derive ten necessary conditions for the stability of dissipative fluids and plasmas. Assuming the validity of the local equilibrium principle, these criteria have been obtained solely from the first and second laws of thermodynamics. The Onsager reciprocity relations have not been invoked, and the author's results are supposed to be valid independent of the choice of the boundary conditions. In the present Comment, in agreement with the general theory established by Glansdorff-Prigogine in 1954 and 1970, we show that there is no variational principle expressing the necessary conditions for the stability of dissipative systems involving convective effects when the system is out of the Onsager region. In particular, we prove that the basic equations constituting the starting point of the analysis of the author, attempting to derive ten necessary conditions for the stability involving magnetohydrodynamical effects, are incorrect and in contradiction with the laws of the thermodynamics of irreversible processes.

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Near a zero-dispersion wavelength, high-order dispersion effects play a central role in a photonic crystal fiber cavity. The study of such effects reveals the existence of stable dissipative dark solitons in the anomalous dispersion regime. Such dark localized structures are not present without high-order dispersion. They consist of dips in the profile of the intensity field. The number of dips and their temporal distribution are determined by the initial conditions. A snaking bifurcation diagram associated with these solutions is constructed.

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We consider a degenerate optical parametric oscillator containing a left-handed material. We show that the inclusion of a left-handed material layer allows for controlling the strength and sign of the diffraction coefficient at either the pump or the signal frequency. Subsequently, we demonstrate the existence of stable dissipative structures without diffraction matching, i.e., without the usual relationship between the diffraction coefficients of the signal and pump fields. Finally, we investigate the size scaling of these light structures with decreasing diffraction strength.

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Localized structures belong to the class of dissipative structures found far from equilibrium. Contributions from the most representative groups working on a various fields of natural science such as biology, chemistry, plant ecology, mathematics, optics, and laser physics are presented. The aim of this issue is to gather specialists from these fields towards a cross-fertilization among these active areas of research and thereby to present an overview of the state of art in the formation and the characterization of dissipative localized structures. Nonlinear optics and laser physics have an important part in this issue because of potential applications in information technology. In particular, localized structures could be used as "bits" for parallel information storage and processing.

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We study the spatiotemporal dynamics of spatially extended nonlinear cavities containing a left-handed material. Such materials, which have a negative index of refraction, have been experimentally demonstrated recently, and allow for novel electromagnetic behavior. We show that the insertion of a left-handed material in an optical resonator allows for controlling the value and the sign of the diffraction coefficient in dispersive Kerr resonators and degenerate optical parametric oscillators. We give an overview of our analytical and numerical studies on the stability and formation of dissipative structures in systems with negative diffraction.