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Seagrasses provide multiple ecosystem services and act as intense carbon sinks in coastal regions around the globe but are threatened by multiple anthropogenic pressures, leading to enhanced seagrass mortality that reflects in the spatial self-organization of the meadows. Spontaneous spatial vegetation patterns appear in such different ecosystems as drylands, peatlands, salt marshes, or seagrass meadows, and the mechanisms behind this phenomenon are still an open question in many cases. Here, we report on the formation of vegetation traveling pulses creating complex spatiotemporal patterns and rings in Mediterranean seagrass meadows. We show that these structures emerge due to an excitable behavior resulting from the coupled dynamics of vegetation and porewater hydrogen sulfide, toxic to seagrass, in the sediment. The resulting spatiotemporal patterns resemble those formed in other physical, chemical, and biological excitable media, but on a much larger scale. Based on theory, we derive a model that reproduces the observed seascapes and predicts the annihilation of these circular structures as they collide, a distinctive feature of excitable pulses. We show also that the patterns in field images and the empirically resolved radial profiles of vegetation density and sediment sulfide concentration across the structures are consistent with predictions from the theoretical model, which shows these structures to have diagnostic value, acting as a harbinger of the terminal state of the seagrass meadows prior to their collapse.

The Eckhaus instability is a secondary instability of nonlinear spatiotemporal patterns in which high-wave-number periodic solutions become unstable against small-wave-number perturbations. Here we show that this instability can take place in Kerr combs corresponding to subcritical Turing patterns upon changes in the laser detuning. The development of the Eckhaus instability leads to the cracking of patterns and a long-lived transient where the peaks of the pattern rearrange in space due to spatial interactions. In the spectral domain, this results in a metastable Kerr comb dynamics with timescales that can be larger than 1 min. This time is, at least, seven orders of magnitude larger than the intracavity photon lifetime and is in sharp contrast with all the transient behaviors reported so far in cavity nonlinear optics that are typically only a few photon lifetimes long (i.e., in the picosecond to the microsecond range). This phenomenology, studied theoretically in the Lugiato-Lefever model and the observed dynamics is compatible with experimental observations in Kerr combs generated in ultra-high-Q whispering-gallery mode resonators., D.G. and Y.K.C. acknowledge support from Project ND-PHOT jointly funded by CSIC and CNRS. D.G. and P.C. acknowledge financial support from Agencia Estatal de Investigación (AEI, Spain) and Fondo Europeo de Desarrollo Regional under Project ESoTECoS, Grant No. FIS2015-63628-C2-1-R (AEI/FEDER,UE) and Agencia Estatal de Investigación through the María de Maeztu Program for Units of Excellence in R&D (Grant No. MDM-2017-0711). Y.K.C. also acknowledges funding from the European Research Council through the Projects NextPhase and Versyt from the Centre National d'Etudes Spatiales (CNES) through Project SHYRO. PPR acknowledges funding from the H2020 Marie Skłodowska-Curie Actions through the Project NOSTER (101023717).

17 pages, 12 figures., We study the scenario in which traveling pulses emerge in a prototypical type-I one-dimensional excitable medium, which exhibits two different routes to excitable behavior, mediated by a homoclinic (saddle-loop) and a saddle-node on the invariant cycle bifurcations. We characterize the region in parameter space in which traveling pulses are stable together with the different bifurcations behind either their destruction or loss of stability. In particular, some of the bifurcations delimiting the stability region have been connected, using singular limits, with the two different scenarios that mediated type-I local excitability. Finally, the existence of traveling pulses has been linked to a drift pitchfork instability of localized steady structures., We acknowledge financial support through grants RTI2018-095441-B-C22 and MDM-2017-0711 funded by MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe. D.R-R. is supported by the Ministry of Universities through the “Pla de Recuperació, Transformació i Resilència” and by the EU (NextGenerationEU), together with the Universitat de les Illes Balears. A.A-i-P. is financed by Czech Science Foundation, Project No. GA19-16066S.

We report experimental observation of subharmonic mode excitation in primary Kerr optical frequency combs generated using crystalline whispering-gallery mode resonators. We show that the subcombs can be controlled and span a single or multiple free spectral ranges around the primary comb modes. In the spatial domain, the resulting multiscale combs correspond to an amplitude modulation of intracavity roll patterns. We perform a theoretical analysis based on eigenvalue decomposition that evidences the mechanism leading to the excitation of these combs.

We consider a general model exhibiting type-I excitability mediated by a homoclinic and a saddle node on the invariant circle bifurcations. We show how the distinct properties of type-I with respect to type-II excitability confer unique features to traveling pulses in excitable media. They inherit the characteristic divergence of type-I excitable trajectories at threshold exhibiting analogous scalings in the spatial thickness of the pulses. Our results pave the way to identify basic underlying mechanisms behind type-I excitable pulses based solely on the characteristics of the pulse.se., Grants RTI2018-095441-B-C22 and MDM-2017-0711 funded by MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe.

We propose a method to analyze the risk of blackouts with high penetration of variable renewable energy sources (VRESs). We consider a model for the long-term evolution of the power grid including propagation of cascading failures, day-to-day fluctuations of renewable generation and moderate use of storage. We analyze grid resilience and stress as VRESs are progressively incorporated. We also evaluate the VRES performance as the average fraction of daily demand covered by renewables.We find that in general, VRES intrinsic variability increases the grid stress and the blackout risk. However, if VRESs are implemented in a distributed way, the spatial spreading of the generation may have a positive effect on grid resilience. As a case study, we analyze the replacement of conventional power plants by solar photovoltaic generation combined with storage in the power grid of the Balearic Islands. We also consider the use of source redundancy and briefly discuss the potential of wind energy. The work of Pere Colet and Damià Gomila was supported in part by the Ministerio de Ciencia e Innovación, Spain, in part by the Agencia Estatal de Investigación (AEI), Spain, in part by the Fondo Europeo de Desarrollo Regional (FEDER, EU) under Grant PACSS RTI2018-093732-B-C22, and in part by the Maria de Maeztu Program for Units of Excellence in Research and Development under Grant MDM-2017-0711. The work of Benjamín A. Carreras and José M. Reynolds-Barredo was supported by the Universidad Carlos III de Madrid, Spain, funded jointly by EU FEDER funds and by the Spanish Government through the National Research Project UNC313-4E-2361, Project ENE2009-12213-C03-03, Project ENE2012-33219, and Project ENE2012-31753 (for providing access to Uranus, a supercomputer cluster).

We investigate the effects of environmental stochastic fluctuations on Kerr optical frequency combs. This spatially extended dynamical system can be accurately studied using the Lugiato-Lefever equation, and we show that when additive noise is accounted for, the correlations of the modal field fluctuations can be determined theoretically. We propose a general theory for the computation of these field fluctuations and correlations, which is successfully compared to numerical simulations., Y. K. C. acknowledges financial support the University of Maryland through the Minta Martin Fellowship

A simplified model of clonal plant growth is formulated, motivated by observations of spatial structures in Posidonia oceanica meadows in the Mediterranean Sea. Two levels of approximation are considered for the scale-dependent feedback terms. Both take into account mortality and clonal, or vegetative, growth as well as competition and facilitation, but the first version is nonlocal in space while the second is local. Study of the two versions of the model in the one-dimensional case reveals that both cases exhibit qualitatively similar behavior (but quantitative differences) and describe the competition between three spatially extended states, the bare soil state, the populated state, and a pattern state, and the associated spatially localized structures. The latter are of two types, holes in the populated state and vegetation patches on bare ground, and are organized within distinct snaking bifurcation diagrams. Fronts between the three extended states are studied and a transition between pushed and pulled fronts identified. Numerical simulations in one spatial dimension are used to determine front speeds and confront the predictions from the marginal stability condition for pulled fronts., DRR, LM, EHG and DG acknowledge financial support from MINECO/AEI/FEDER through project SuMaEco (RTI2018-095441-B-C22) and through the María de Maeztu Program for Units of Excellence in R&D (MDM- 2017-0711). DRR also acknowledges the fellowship BES-2016-076264 under FPI program of MINECO, Spain. The work of EK was supported in part by National Science Foundation under grant DMS-1908891, and by a visiting position at IFISC funded by the University of the Balearic Islands.

Vegetation patterns, a natural phenomenon observed worldwide, are typically driven by spatially distributed feedback. However, the spatial colonization mechanisms of clonal plants, driven by the growth of a rhizome, are usually not considered in prototypical models. Here we propose a general equation for the vegetation density that includes all main clonal-growth features as well as the essential ingredients leading to spatial self-organization. This generic model reproduces the phase diagram of a fully detailed model of clonal growth. The relation of each term of the model with the mechanisms of clonal growth is discussed., We acknowledge financial support from FEDER/Ministerio de Ciencia, Innovación y Universidades Agencia Estatal de Investigación/ SuMaEco Project (RTI2018-095441-B-C22) and the María de Maeztu Program for Units of Excellence in R&D (No. MDM-2017-0711). D.R.-R. also acknowledges the Fellowship No. BES-2016-076264 under the FPI program of MINECO, Spain.

The propagation of failures and blackouts in electric networks is a complex problem. Typical models, such as the ORNL-PSerc-Alaska (OPA), are based on a combination of fast and a slow dynamics. The first describes the cascading failures while the second the grid evolution though line and generation upgrades as well as demand growth, all taking place in time scales from days to years. The growing integration of renewable energy sources, whose power fluctuates in time scales from seconds to hours, together with the increase in demand, which also present fast fluctuations, require the incorporation of distributed methods of control in the demand side to avoid the high cost of ordinary control in conventional power plants. In this work, we extend the OPA model to include fluctuations in the demand at time scales of the order of minutes, intraday demand variations and the effect of demand control. We find that demand control effectively reduces the number of blackouts without increasing the probability of large-scale events., The authors E.B.T.T., D.G. and P.C. acknowledge fund-ing from Ministerio de Ciencia e Innovación (Spain), theAgencia Estatal de Investigación (AEI, Spain), and the Fondo Europeo de Desarrollo Regional (FEDER, EU) under grant PACSS (RTI2018-093732-B-C22) and the Maria de Maeztu program for Units of Excellence in R&D (MDM-2017-0711).E.B.T.T. also acknowledges the fellowship FIS2015-63628-CZ-Z-R under the FPI program of AEI and MINEICO, Spain.B.A.C. and J.M.R.-B. acknowledge access to Uranus, a super-computer cluster located at Universidad Carlos III de Madrid(Spain).

Clonal growth plants are abundant in both terrestrial and marine ecosystems. Some marine species are particularly important since they provide essential ecosystem services in the shores of all continents except Antarctica. For the appropriate modelization of clonal growth the discretization of the growth direction angle has to be carefully treated to correctly describe the dynamics of the meadow. Specifically, determining the minimum number of growth directions is important to maximize the efficiency of numerical simulations. We show that the presence of neutral modes in the growth direction angle tends to make the distribution of the growth directions uniform in the presence of noise, allowing us to choose the minimal number of discrete angles compatible with the branching. We also show that the formation of spatial patterns induces small differences in the population density within these different growth directions., We acknowledge financial support from Agencia Estatal de Investigación (AEI, Spain) and Fondo Europeo de Desarrollo Regional under Project SuMaEco Grant No. RTI2018-095441-B-C22 (AEI/FEDER,UE) and the Spanish State Research Agency, through the Maria de Maeztu Program for units of Excellence in R&D (MDM-2017-0711). D.R-R. also acknowledges the fellowship BES-2016-076264 under the FPI program of MINECO, Spain.

We study the stability and bifurcation structure of spatially extended patterns arising in nonlin- ear optical resonators with a Kerr-type nonlinearity and anomalous group velocity dispersion, as described by the Lugiato-Lefever equation. While there exists a one-parameter family of patterns with different wavelengths, we focus our attention on the pattern with critical wave number k c arising from the modulational instability of the homogeneous state. We find that the branch of solutions associated with this pattern connects to a branch of patterns with wave number $2k_c$ . This next branch also connects to a branch of patterns with double wave number, this time $4k_c$ , and this process repeats through a series of 2:1 spatial resonances. For values of the detuning parameter approaching $\theta = 2$ from below the critical wave number $k_c$ approaches zero and this bifurcation structure is related to the foliated snaking bifurcation structure organizing spatially localized bright solitons. Secondary bifurcations that these patterns undergo and the resulting temporal dynamics are also studied., Comment: 13 pages, 13 figures

The increase of the electric demand and the progressive integration of renewable energy sources threatens the stability of the power grid. To solve this issue, several methods have been proposed to control the demand side instead of increasing the spinning reserve in the supply. Here we focus on dynamic demand control (DDC), a method in which smart devices can autonomously delay its scheduled operation if the electric frequency is outside a suitable range. While typical control schemes can effectively reduce small and medium size frequency fluctuations, the probability of large demand peaks, and hence large frequency fluctuation, may increase due to the need of recovering pending tasks. Although these events are rare they can potentially trigger a failure of the system and therefore strategies to avoid them need to be addressed. In this work we introduce a method including communication among DDC devices belonging to a given group, such that they can coordinate opposite actions to keep the group demand more stable. We show that, with minimal information exchange, our method reduces the amount of pending tasks by a factor 10 while large frequency fluctuations are significantly reduced or even completely avoided., We acknowledge financial support from Agencia Estatal de Investigación (AEI, Spain) and Fondo Europeo de Desarrollo Regional under Project ESoTECoS, Grant Nos.: FIS2015-63628-C2-1-R (AEI/FEDER,UE) and FIS2015-63628-C2-2-R (AEI/FEDER,UE) and Agencia Estatal de Investigación through María de Maestu Program for Units of Excellence in R&D (MDM-2017-0711). E.B.T.T. also acknowledges the fellowship FIS2015-63628-CZ-Z-R under the FPI program of AEI, Spain.

Short-scale interactions yield large-scale vegetation patterns that, in turn, shape ecosystem function across landscapes. Fairy circles, which are circular patches bare of vegetation within otherwise continuous landscapes, are characteristic features of semiarid grasslands. We report the occurrence of submarine fairy circle seascapes in seagrass meadows and propose a simple model that reproduces the diversity of seascapes observed in these ecosystems as emerging from plant interactions within the meadow. These seascapes include two extreme cases, a continuous meadow and a bare landscape, along with intermediate states that range from the occurrence of persistent but isolated fairy circles, or solitons, to seascapes with multiple fairy circles, banded vegetation, and “leopard skin” patterns consisting of bare seascapes dotted with plant patches. The model predicts that these intermediate seascapes extending across kilometers emerge as a consequence of local demographic imbalances along with facilitative and competitive interactions among the plants with a characteristic spatial scale of 20 to 30 m, consistent with known drivers of seagrass performance. The model, which can be extended to clonal growth plants in other landscapes showing fairy rings, reveals that the different seascapes observed hold diagnostic power as to the proximity of seagrass meadows to extinction points that can be used to identify ecosystems at risks., D.R.-R., D.G., T.S., E.H.-G., and N.M. acknowledge financial support from AEI/FEDER [Agencia Estatal de Investigación/Fondo Europeo de Desarrollo Regional, European Union (EU)] (FIS2015-63628-C2-1-R, FIS2015-63628-C2-2-R, and CGL2015-71809-P). C.M.D. was supported by King Abdullah University of Science and Technology through the baseline funding., We acknowledge support by the CSIC Open Access Publication Initiative through its Unit of Information Resources for Research (URICI).

Topical Issue: Theory and applications of the Lugiato-Lefever Equation., Bound states, also called soliton molecules, can form as a result of the interaction between individual solitons. This interaction is mediated through the tails of each soliton that overlap with one another. When such soliton tails have spatial oscillations, locking or pinning between two solitons can occur at fixed distances related with the wavelength of these oscillations, thus forming a bound state. In this work, we study the formation and stability of various types of bound states in the Lugiato-Lefever equation by computing their interaction potential and by analyzing the properties of the oscillatory tails. Moreover, we study the effect of higher order dispersion and noise in the pump intensity on the dynamics of bound states. In doing so, we reveal that perturbations to the Lugiato-Lefever equation that maintain reversibility, such as fourth order dispersion, lead to bound states that tend to separate from one another in time when noise is added. This separation force is determined by the shape of the envelope of the interaction potential, as well as an additional Brownian ratchet effect. In systems with broken reversibility, such as third order dispersion, this ratchet effect continues to push solitons within a bound state apart. However, the force generated by the envelope of the potential is now such that it pushes the solitons towards each other, leading to a null net drift of the solitons., We acknowledge support from the Research Foundation-Flanders (FWO-Vlaanderen) (P.P.-R.), the Belgian Science Policy Office (BelSPO) under Grant No. IAP 7-35, the Research Council of the Vrije Universiteit Brussel, and Agencia Estatal de Investigación (AEI, Spain) and Fondo Europeo de Desarrollo Region al under Project ESoTECoS FIS2015-63628-C2-1-R (AEI/FEDER,UE) (D.G. and P.C.).

Power grid frequency control is a demanding task requiring expensive idle power plants to adapt the supply to the fluctuating demand. An alternative approach is controlling the demand side in such a way that certain appliances modify their operation to adapt to the power availability. This is especially important to achieve a high penetration of renewable energy sources. A number of methods to manage the demand side have been proposed. In this work we focus on dynamic demand control (DDC), where smart appliances can delay their switchings depending on the frequency of the system. We introduce a simple model to study the effects of DDC on the frequency of the power grid. The model includes the power plant equations, a stochastic model for the demand that reproduces, adjusting a single parameter, the statistical properties of frequency fluctuations measured experimentally, and a generic DDC protocol. We find that DDC can reduce small and medium-size fluctuations but it can also increase the probability of observing large frequency peaks due to the necessity of recovering pending task. We also conclude that a deployment of DDC around 30–40% already allows a significant reduction of the fluctuations while keeping the number of pending tasks low., We acknowledge helpful discussions with B. A. Carreras and financial support from Agencia Estatal de Investigación (AEI, Spain) and Fondo Europeo de Desarrollo Regional under Project No. ESoTECoS Grants No. FIS2015-63628-C2-1-R (AEI/FEDER,UE) and No. FIS2015-63628-C2-2-R (AEI/FEDER,UE). E.B.T.-T. also acknowledges the fellowship FIS2015-63628-CZ-Z-R under the FPI program of MINEICO, Spain.

Using the Lugiato-Lefever model, we analyze the effects of third-order chromatic dispersion on the existence and stability of dark- and bright-soliton Kerr frequency combs in the normal dispersion regime. While in the absence of third-order dispersion only dark solitons exist over an extended parameter range, we find that third-order dispersion allows for stable dark and bright solitons to coexist. Reversibility is broken and the shape of the switching waves connecting the top and bottom homogeneous solutions is modified. Bright solitons come into existence thanks to the generation of oscillations in the switching-wave profiles. Temporal oscillatory instabilities of dark solitons are suppressed in the presence of sufficiently strong third-order dispersion, while bright solitons are never found to oscillate in time. As a result of third-order dispersion both bright and dark solitons are found to move with a velocity that depends on their width., We acknowledge support from the Research Foundation–Flanders (FWO-Vlaanderen) (P.P.-R.), the Belgian Science Policy Office (BelSPO) under Grant No. IAP 7-35, the Research Council of the Vrije Universiteit Brussel, and Ministerio de Economía y Competitividad (Spain) and Fondo Europeo de Desarrollo Regional under Project ESoTECoS FIS2015-63628-C2-1-R (MINECO/FEDER) (D.G.).

Spatially extended systems can support local transient excitations in which just a part of the system is excited. The mechanisms reported so far are local excitability and excitation of a localized structure. Here we introduce an alternative mechanism based on the coexistence of two homogeneous stable states and spatial coupling. We show the existence of a threshold for perturbations of the homogeneous state. Subthreshold perturbations decay exponentially. Superthreshold perturbations induce the emergence of a long-lived structure formed by two back to back fronts that join the two homogeneous states. While in typical excitability the trajectory follows the remnants of a limit cycle, here reinjection is provided by front interaction, such that fronts slowly approach each other until eventually annihilating. This front-mediated mechanism shows that extended systems with no oscillatory regimes can display excitability., We acknowledge individual support (P.P.-R.) and project support (L.G.) by the Research Foundation - Flanders (Fonds Wetenschappelijk Onderzoek, FWO-Vlaanderen), by the IAP, and by the research council of the Vrije Universiteit Brussel (VUB). We also acknowledge financial support from Agencia Estatal de Investigación (AEI, Spain) and Fondo Europeo de Desarrollo Regional under Project ESoTECoS FIS2015-63628-C2-1-R (AEI/FEDER,UE).

We analyze dark pulse Kerr frequency combs in optical resonators with normal group-velocity dispersion using the Lugiato−Lefever model. We show that in the time domain the combs correspond to interlocked switching waves between the upper and lower homogeneous states, and explain how this fact accounts for many of their experimentally observed properties. Modulational instability does not play any role in their existence. We provide a detailed map indicating for which parameters stable dark pulse Kerr combs can be found, and how they are destabilized for increasing values of frequency detuning., Research Foundation–Flanders (FWO-Vlaanderen); KU Leuven Junior Mobility Programme; Belgian Science Policy Office (BelSPO) (IAP 7-35); Vrije Universiteit Brussel, Research Council; MINECO and FEDER (Intense@Cosyp (FIS2012-30634)); National Science Foundation (NSF) (DMS-1211953); Marsden Fund of the Royal Society of New Zealand.

In this work we present a detailed analysis of bifurcation structures of cavity solitons (CSs) and determine the different dynamical regimes in the Lugiato-Lefever (LL) equation in the presence of anomalous and normal chromatic dispersion regimes. Such an analysis has been shown to also increase our understanding of frequency combs (FCs). A FC consists in a set of equidistant spectral lines that can be used to measure light frequencies and time intervals more easily and precisely than ever before. Due to the duality between CSs in microcavities and FCs, we can gain information about the behavior of FCs by analyzing the dynamics of CSs. In the anomalous dispersion case bright CSs are organized in what is known as a homoclinic snaking bifurcation structure. In contrast, in the normal dispersion regime dark CSs are organized differently, in a structure known as collapsing snaking. Despite the differences in bifurcation scenarios, both types of CSs present similar temporal instabilities. © (2016) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.

6 pages, 8 figures.-- PDF pre-print., We study the effects of noise on excitable DS found on nonlinear Kerr cavities, showing that the system exhibits coherence resonance, characterized by a maximum degree of regularity for intermediate noise intensities. This behavior is observed for two different ways of applying noise: an additive white uncorrelated spatio-temporal noise and including fluctuations in the intensity of an addressing beam., We acknowledge financial support from MICINN (Spain) and FEDER through project FISICOS (FIS2007-60327).

This chapter presents the general theory about how different dissipative soliton (DS) dynamics are induced by inhomogeneities and drift. For this purpose, the prototypical Swift-Hohenberg equation (SHE) is used. The chapter focuses on the mechanism that leads to excitable dynamics. It discusses the modeling of fiber cavities in terms of the Lugiato-Lefever model, and describes the effect of periodic pumping in ring cavities. In fiber cavities and microresonators, inhomogeneities and drift are unavoidable due to imperfections in the fabrication process, material properties, and higher-order chromatic light dispersion. In the context of fiber cavities and microresonators, periodic boundary conditions lead to the emission of a train of solitons from the inhomogeneity. These trains of solitons lie at the basis of Kerr frequency comb generation. Therefore, the dynamical instabilities of cavity solitons induced by imperfections of themicroresonator can be particularly relevant for applications relying on stable frequency combs.

We theoretically analyze dark pulse Kerr frequency combs in normal dispersion microresonators. A wide range of dark pulses of different widths are found to coexist, and can be described as interlocked switching waves.

We theoretically analyze dark pulse Kerr frequency combs in normal dispersion microresonators. A wide range of dark pulses of different widths are found to coexist, and can be described as interlocked switching waves.

We have reported in Phys. Rev. Lett. 110, 064103 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064103 that in systems which otherwise do not show oscillatory dynamics, the interplay between pinning to a defect and pulling by drift allows the system to exhibit excitability and oscillations. Here we build on this work and present a detailed bifurcation analysis of the various dynamical instabilities that result from the competition between a pulling force generated by the drift and a pinning of the solitons to spatial defects. We show that oscillatory and excitable dynamics of dissipative solitons find their origin in multiple codimension-2 bifurcation points. Moreover, we demonstrate that the mechanisms leading to these dynamical regimes are generic for any system admitting dissipative solitons., This research was supported by the Research Foundation–Flanders (FWO-Vlaanderen), by the Junior Mobility Programme (JuMo) at KU Leuven, by the Belgian Science Policy Office (BelSPO) under Grant No. IAP 7-35, by the Research Council of the Vrije Universiteit Brussel, by the Ministerio de Economía y Competitividad (Spain) and Fondo Europeo de Desarrollo Regional under Grants Intense@Cosyp (FIS2012-30634) and TripHop (TEC2012-36335).

The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understanding frequency comb generation in microresonators is emphasized., This research was supported by the Research Foundation–Flanders (FWO-Vlaanderen) (P.P. and L.G.), by the Junior Mobility Programme (JuMo) of the KU Leuven (L.G.), by the Belgian Science Policy Office (BelSPO) under Grant No. IAP 7-35 (P.P. and L.G.), by the Research Council of the Vrije Universiteit Brussel (P.P. and L.G.), by the Spanish MINECO and FEDER under Grant Intense@Cosyp (FIS2012-30634) (D.G.), and by the National Science Foundation under Grant No. DMS-1211953 (E.K.).

In this paper we analyze in detail the structure of the phase space of a reversible dynamical system describing the stationary solutions of a model for a nonlinear optical cavity. We compare our results with the general picture described in [P.D. Woods, A.R. Champneys, Physica D 129 (1999) 147; P. Coullet, C. Riera, C. Tresser, Phys. Rev. Lett. 84 (2000) 3069] and find that the stable and unstable manifolds of homogeneous and patterned solutions present a much higher level of complexity than predicted, including the existence of additional localized solutions and fronts. This extra complexity arises due to homoclinic and heteroclinic intersections of the invariant manifolds of low-amplitude periodic solutions, and to the fact that these periodic solutions together with the high-amplitude ones constitute a one-parameter family generating a closed line on the symmetry plane.

5 pages.-- Final full-text version of the paper available at: http://dx.doi.org/10.2529/PIERS060907134206., We characterize a scenario where localized structures in nonlinear optical cavities display an oscillatory behavior which becomes unstable leading to an excitable regime. Excitability emerges from spatial dependence since the system locally is not excitable. We show the existence of different mechanisms leading to excitability depending on the profile of the pump field., We acknowledge financial support from MEC (Spain) and FEDER: Grants BFM2001-0341-C02-02, FIS2004-00953 (CONOCE2) and FIS2004-05073-C04-03. AJ acknowledges financial support from the MEC.

6 pages, 10 figures.-- PACS nrs.: 42.65.Sf; 05.65.+b; 45.70.Qj., The universality of two mechanisms of motion of patterns in nonlinear optics is demonstrated. In the first one, two-dimensional disordered patterns are shown to scroll with constant velocity after transients are discarded. In the second case, pattern motion is induced by background modulations. Spatially periodic patterns lock to maxima or minima of the underlying modulation depending on its wavevector., We acknowledge support from the European projects QUANTIM and FunFACS and from EPSRC. AJS and GLO acknowledge support from SGI. GLO thanks the Royal Society–Leverhulme Trust and Centro Studi Dinamiche Complesse, University of Florence, Sesto Fiorentino, Italy.

6 pages, 7 figures., The existence and stability properties of localized structures in nonlinear optical cavities with slightly non-equivalent homogeneous solutions and displaying a modulational instability of flat fronts are investigated. We present a new type of stable localized structures in the regime of formation of labyrinthine patterns based on the balance between the curvature and the asymmetry effects., We acknowledge financial support from the European Commission project QUANTIM (IST-2000-26019). D Gomila acknowledges financial support from EPSRC (Grant No. GR/S28600/01). P. Colet and M. San Miguel acknowledge financial support from MCyT (Spain) and FEDER under projects BFM2000-1108 (CONOCE) and BFM2001-0341-C02-02 (SINFIBIO). G-L Oppo acknowledges support form SGI, the Royal Society Leverhulme Trust, the PCRI consortium of the University of Strathclyde and EPSRC (Grant No. GR/R04096).

Optical frequency combs can be used to measure light frequencies and time intervals more easily and precisely than ever before, opening a large avenue for applications. Traditional frequency combs are usually associated with trains of evenly spaced, very short pulses. More recently, a new generation of comb sources has been demonstrated in compact high-Q optical microresonators with a Kerr nonlinearity pumped by continuous-wave laser light. These combs are now referred to as Kerr frequency combs and have attracted a lot of interest in the last few years. Kerr frequency combs can be modeled in a way that is strongly reminiscent of temporal cavity solitons (CSs) in nonlinear cavities. Temporal CSs have been experimentally studied in fiber resonators and their description is based on a now classical equation, the Lugiato-Lefever equation, that describes pattern formation in optical systems. In this work, we first perform a theoretical study of the correspondence between the CSs and patterns with frequency combs. It is known that the CSs appear in reversible systems that present bistability between a pattern and a homogeneous steady state through what it is called a homoclinic snaking structure. In this snaking region, single and multi-peak CSs coexist with patterns and homogeneous solutions, creating a largely multistable landscape. We study the changes of the homoclinic snaking for different parameter regimes in the Lugiato-Lefever equation and determine the stability and shape of the frequency combs through comparison with the underlying CSs and patterns. Secondly, we include third order dispersion in the system and study its effect on the multistable snaking structure. For high dispersion strengths the CS structures and the corresponding Kerr frequency combs disappear.

The Lugiato-Lefever equation (LLE) has been extensively studied since its derivation in 1987, when this meanfield model was introduced to describe nonlinear optical cavities. The LLE was originally derived to describe a ring cavity or a Fabry-Perot resonator with a transverse spatial extension and partially filled with a nonlinear medium but it has also been shown to be applicable to other types of cavities, such as fiber resonators and microresonators. Depending on the parameters used, the LLE can present a monostable or bistable input-output response curve. A large number of theoretical studies have been done in the monostable regime, but the bistable regime has remained widely unexplored. One of the reasons for this was that previous experimental setups were not able to works in such regimes of the parameter space. Nowadays the possibility of reaching such parameter regimes experimentally has renewed the interest in the LLE. In this contribution, we present an in-depth theoretical study of the different dynamical regimes that can appear in parameter space, focusing on the dynamics of localized solutions, also known as cavity solitons (CSs). We show that time-periodic oscillations of a 1D CS appear naturally in a broad region of parameter space. More than this oscillatory regime, which has been recently demonstrated experimentally,1 we theoretically report on several kinds of chaotic dynamics. We show that the existence of CSs and their dynamics is related with the spatial dynamics of the system and with the presence of a codimension-2 point known as a Fold-Hopf bifurcation point. These dynamical regimes can become accessible by using devices such as microresonators, for instance widely used for creating optical frequency combs.

It has been recently uncovered that coherent structures in microresonators such as cavity solitons and patterns are intimately related to Kerr frequency combs. In this work, we present a general analysis of the regions of existence and stability of cavity solitons and patterns in the Lugiato-Lefever equation, a mean-field model that finds applications in many different nonlinear optical cavities. We demonstrate that the rich dynamics and coexistence of multiple solutions in the Lugiato-Lefever equation are of key importance to understanding frequency comb generation. A detailed map of how and where to target stable Kerr frequency combs in the parameter space defined by the frequency detuning and the pump power is provided. Moreover, the work presented also includes the organization of various dynamical regimes in terms of bifurcation points of higher codimension in regions of parameter space that were previously unexplored in the Lugiato-Lefever equation. We discuss different dynamical instabilities such as oscillations and chaotic regimes., This research was supported by the Research Foundation-Flanders (FWO), by the Spanish MINECO, and FEDER under Grants FISICOS (Grant No. FIS2007-60327) and INTENSE@COSYP (Grant No. FIS2012-30634), by Comunitat Autonoma de les Illes Balears, by the Research Council of the Vrije Universiteit Brussel (VUB), and by the Belgian Science Policy Office (BelSPO) under Grant No. IAP 7-35. S. Coen also acknowledges the support of the Marsden Fund of the Royal Society of New Zealand.

A dynamical model of the Swift-Hohenberg type is proposed to describe the formation of twelvefold quasipattern as observed, for instance, in optical systems. The model incorporates the general mechanisms leading to quasipattern formation and does not need external forcing to generate them. Besides quadratic nonlinearities, the model takes into account an angular dependence of the nonlinear couplings between spatial modes with different orientations. Furthermore, the marginal stability curve presents other local minima than the one corresponding to critical modes, as usual in optical systems. Quasipatterns form when one of these secondary minima may be associated with harmonics built on pairs of critical modes. The model is analyzed numerically and in the framework of amplitude equations. The results confirm the importance of harmonics to stabilize quasipatterns and assess the applicability of the model to other systems with similar generic properties. © 2014 American Physical Society., We acknowledge useful discussions with P. Colet, and financial support from FEDER and MINECO (Spain), through Grant No. FIS2012-30634 INTENSE@COSYP, and from Comunitat Autónoma de les Illes Balears.