Bifurcation structure of cavity soliton dynamics in a VCSEL with saturable absorber and time-delayed feedback

We consider a wide-aperture surface-emitting laser with a saturable absorber section subjected to time-delayed feedback. We adopt the mean-field approach assuming a single longitudinal mode operation of the solitary VCSEL. We investigate cavity soliton dynamics under the effect of time- delayed feedback in a self-imaging configuration where diffraction in the external cavity is negligible. Using bifurcation analysis, direct numerical simulations and numerical path continuation methods, we identify the possible bifurcations and map them in a plane of feedback parameters. We show that for both the homogeneous and localized stationary lasing solutions in one spatial dimension the time-delayed feedback induces complex spatiotemporal dynamics, in particular a period doubling route to chaos, quasiperiodic oscillations and multistability of the stationary solutions

Extended patchy ecosystems may increase their total biomass through self-replication

Patches of vegetation consist of dense clusters of shrubs, grass, or trees, often found to be circular characteristic size, defined by the properties of the vegetation and terrain. Therefore, vegetation patches can be interpreted as localized structures. Previous findings have shown that such localized structures can self-replicate in a binary fashion, where a single vegetation patch elongates and divides into two new patches. Here, we extend these previous results by considering the more general case, where the plants interact non-locally, this extension adds an extra level of complexity and shrinks the gap between the model and real ecosystems, where it is known that the plant-to-plant competition through roots and above-ground facilitating interactions have non-local effects, i.e. they extend further away than the nearest neighbor distance. Through numerical simulations, we show that for a moderate level of aridity, a transition from a single patch to periodic pattern occurs. Moreover, for large values of the hydric stress, we predict an opposing route to the formation of periodic patterns, where a homogeneous cover of vegetation may decay to spot-like patterns. The evolution of the biomass of vegetation patches can be used as an indicator of the state of an ecosystem, this allows to distinguish if a system is in a self-replicating or decaying dynamics. In an attempt to relate the theoretical predictions to real ecosystems, we analyze landscapes in Zambia and Mozambique, where vegetation forms patches of tens of meters in diameter. We show that the properties of the patches together with their spatial distributions are consistent with the self-organization hypothesis. We argue that the characteristics of the observed landscapes may be a consequence of patch self-replication, however, detailed field and temporal data is fundamental to assess the real state of the ecosystems.Comment: 38 pages, 12 figures, 1 tabl

Stabilization of localized structures by inhomogeneous injection in Kerr resonators

We consider the formation of temporal localized structures or Kerr comb generation in a microresonator with inhomogeneities. We show that the introduction of even a small inhomogeneity in the injected beam widens the stability region of localized solutions. The homoclinic snaking bifurcation associated with the formation of localized structures and clusters of them with decaying oscillatory tails is constructed. Furthermore, the inhomogeneity allows not only to control the position of localized solutions, but strongly affects their stability domains. In particular, a new stability domain of a single peak localized structure appears outside of the region of multistability between multiple peaks of localized states. We identify a regime of larger detuning, where localized structures do not exhibit a snaking behavior. In this regime, the effect of inhomogeneities on localized solutions is far more complex: they can act either attracting or repelling. We identify the pitchfork bifurcation responsible for this transition. Finally, we use a potential well approach to determine the force exerted by the inhomogeneity and summarize with a full analysis of the parameter regime where localized structures and therefore Kerr comb generation exist and analyze how this regime changes in the presence of an inhomogeneity

Dissipative soliton interaction in Kerr resonators with high-order dispersion

We consider an optical resonator containing a photonic crystal fiber and driven coherently by an injected beam. This device is described by a generalized Lugiato--Lefever equation with fourth order dispersion We use an asymptotic approach to derive interaction equations governing the slow time evolution of the coordinates of two interacting dissipative solitons. We show that Cherenkov radiation induced by positive fourth-order dispersion leads to a strong increase of the interaction force between the solitons. As a consequence, large number of equidistant soliton bound states in the phase space of the interaction equations can be stabilized. We show that the presence of even small spectral filtering not only dampens the Cherenkov radiation at the soliton tails and reduces the interaction strength, but can also affect the bound state stability

Relative stability of multipeak localized patterns

We study relative stability properties of different clusters of closely packed one- and two-dimensional localized peaks of the Swift-Hohenberg equation. We demonstrate the existence of a 'spatial Maxwell' point where clusters are almost equally stable, irrespective of the number of pes involved. Above (below) the Maxwell point, clusters become more (less) stable with the increase of the number of peaks

Spontaneous motion of localized structures induced by parity symmetry transition

We consider a paradigmatic nonvariational scalar Swift-Hohenberg equation that describes short wavenumber or large wavelength pattern forming systems. This work unveils evidence of the transition from stable stationary to moving localized structures in one spatial dimension as a result of a parity breaking instability. This behavior is attributed to the nonvariational character of the model. We show that the nature of this transition is supercritical. We characterize analytically and numerically this bifurcation scenario from which emerges asymmetric moving localized structures. A generalization for two-dimensional settings is discussed.Comment: 7 pages, 6 figure

Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction in water-in-oil microemulsion

In the limit of large diffusivity ratio, spot-like solutions in the two-dimensional Belousov-Zhabotinski reaction in water-in-oil microemulsion are studied. It is shown analytically that such spots undergo an instability as the diffusivity ratio is decreased. An instability threshold is derived. For spots of small radius, it is shown that this instability leads to a spot splitting into precisely two spots. For larger spots, it leads to deformation, fingering patterns and space-filling curves. Numerical simulations are shown to be in close agreement with the analytical predictions.Comment: To appear, PR

Effect of Cherenkov radiation on localized states interaction

We study theoretically the interaction of temporal localized states in all fiber cavities and microresonator-based optical frequency comb generators. We show that Cherenkov radiation emitted in the presence of third order dispersion breaks the symmetry of the localized structures interaction and greatly enlarges their interaction range thus facilitating the experimental observation of the dissipative soliton bound states. Analytical derivation of the reduced equations governing slow time evolution of the positions of two interacting localized states in a generalized Lugiato-Lefever model with the third order dispersion term is performed. Numerical solutions of the model equation are in close agreement with analytical predictions