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Akhmediev, N., Kibler, B., Baronio, F., Belic, M., Zhong, W. P., Zhang, Y. Q., et al. (2016). Roadmap on optical rogue waves and extreme events. J. Opt., 18(6), 37 pp.
Abstract: 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 14). 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.
Keywords: rogue waves; extreme events; nonlinear optics

Barra, F., Lund, F., Mujica, N., & Rica, S. (2012). Shear modulus of an elastic solid under external pressure as a function of temperature: The case of helium. Phys. Rev. B, 85(6), 6 pp.
Abstract: The energy of a dislocation loop in a continuum elastic solid under pressure is considered within the framework of classical mechanics. For a circular loop, this is a function with a maximum at pressures that are well within reach of experimental conditions for solid helium, suggesting, in this case, that dislocation loops can be generated by a pressureassisted thermally activated process. It is also pointed out that pinned dislocation segments can alter the shear response of solid helium by an amount consistent with current measurements, without any unpinning.

Baudin, K., Fusaro, A., Krupa, K., Garnier, J., Rica, S., Millot, G., et al. (2020). Classical RayleighJeans Condensation of Light Waves: Observation and Thermodynamic Characterization. Phys. Rev. Lett., 125(24), 244101.
Abstract: Theoretical studies on wave turbulence predict that a purely classical system of random waves can exhibit a process of condensation, which originates in the singularity of the RayleighJeans equilibrium distribution. We report the experimental observation of the transition to condensation of classical optical waves propagating in a multimode fiber, i.e., in a conservative Hamiltonian system without thermal heat bath. In contrast to conventional selforganization processes featured by the nonequilibrium formation of nonlinear coherent structures (solitons, vortices, ...), here the selforganization originates in the equilibrium RayleighJeans statistics of classical waves. The experimental results show that the chemical potential reaches the lowest energy level at the transition to condensation, which leads to the macroscopic population of the fundamental mode of the optical fiber. The nearfield and farfield measurements of the condensate fraction across the transition to condensation are in quantitative agreement with the RayleighJeans theory. The thermodynamics of classical wave condensation reveals that the heat capacity takes a constant value in the condensed state and tends to vanish above the transition in the normal state. Our experiments provide the first demonstration of a coherent phenomenon of selforganization that is exclusively driven by optical thermalization toward the RayleighJeans equilibrium.

Canals, C., Goles, E., Mascareno, A., Rica, S., & Ruz, G. A. (2018). School Choice in a Market Environment: Individual versus Social Expectations. Complexity, 3793095, 11 pp.
Abstract: School choice is a key factor connecting personal preferences (beliefs, desires, and needs) and school offer in education markets. While it is assumed that preferences are highly individualistic forms of expectations by means of which parents select schools satisfying their internal moral standards, this paper argues that a better matching between parental preferences and school offer is achieved when individuals take into account their relevant network vicinity, thereby constructing social expectations regarding school choice. We develop two related models (individual expectations and social expectations) and prove that they are driven by a Lyapunov function, obtaining that both models converge to fixed points. Also, we assess their performance by conducting computational simulations. While the individual expectations model shows a probabilistic transition and a critical threshold below which preferences concentrate in a few schools and a significant amount of students is left unattended by the school offer, the social expectations model presents a smooth dynamics in which most of the schools have students all the time and no students are left out. We discuss our results considering key topics of the empirical research on school choice in educational market environments and conclude that social expectations contribute to improve information and lead to a better matching between school offer and parental preferences.

Clerc, M. G., Rica, S., & Tredicce, J. (2011). Instabilities and Nonequilibrium Structures. On the occasion of the 60th birthday of Pierre Coullet. Eur. Phys. J. D, 62(1), 1–4. 
Cortez, V., Medina, P., Goles, E., Zarama, R., & Rica, S. (2015). Attractors, statistics and fluctuations of the dynamics of the Schelling's model for social segregation. Eur. Phys. J. B, 88(1), 12 pp.
Abstract: Statistical properties, fluctuations and probabilistic arguments are shown to explain the robust dynamics of the Schelling's social segregation model. With the aid of probability density functions we characterize the attractors for multiple external parameters and conditions. We discuss the role of the initial states and we show that, indeed, the system evolves towards well defined attractors. Finally, we provide probabilistic arguments to explain quantitatively the observed behavior.

Domic, N. G., Goles, E., & Rica, S. (2011). Dynamics and complexity of the Schelling segregation model. Phys. Rev. E, 83(5), 13 pp.
Abstract: In this paper we consider the Schelling social segregation model for two different populations. In Schelling's model, segregation appears as a consequence of discrimination, measured by the local difference between two populations. For that, the model defines a tolerance criterion on the neighborhood of an individual, indicating wether the individual is able to move to a new place or not. Next, the model chooses which of the available unhappy individuals really moves. In our work, we study the patterns generated by the dynamical evolution of the Schelling model in terms of various parameters or the initial condition, such as the size of the neighborhood of an inhabitant, the tolerance, and the initial number of individuals. As a general rule we observe that segregation patterns minimize the interface of zones of different people. In this context we introduce an energy functional associated with the configuration which is a strictly decreasing function for the tolerant people case. Moreover, as far as we know, we are the first to notice that in the case of a nonstrictlydecreasing energy functional, the system may segregate very efficiently.

During, G., Josserand, C., Krstulovic, G., & Rica, S. (2019). Strong turbulence for vibrating plates: Emergence of a Kolmogorov spectrum. Phys. Rev. Fluids, 4(6), 12 pp.
Abstract: In fluid turbulence, energy is transferred from one scale to another by an energy cascade that depends only on the energydissipation rate. It leads by dimensional arguments to the Kolmogorov 1941 (K41) spectrum. Here we show that the normal modes of vibrations in elastic plates also manifest an energy cascade with the same K41 spectrum in the fully nonlinear regime. In particular, we observe different patterns in the elastic deformations such as folds, developable cones, and even more complex stretching structures, in analogy with spots, swirls, vortices, and other structures in hydrodynamic turbulence. We show that the energy cascade is dominated by the kinetic contribution and that the stretching energy is at thermodynamical equilibrium. We characterize this energy cascade, the validity of the constant energydissipation rate over the scales. Finally, we discuss the role of intermittency using the correlation functions that exhibit anomalous exponents.

During, G., Josserand, C., & Rica, S. (2015). Selfsimilar formation of an inverse cascade in vibrating elastic plates. Phys. Rev. E, 91(5), 10 pp.
Abstract: The dynamics of random weakly nonlinear waves is studied in the framework of vibrating thin elastic plates. Although it has been previously predicted that no stationary inverse cascade of constant wave action flux could exist in the framework of wave turbulence for elastic plates, we present substantial evidence of the existence of a timedependent inverse cascade, opening up the possibility of selforganization for a larger class of systems. This inverse cascade transports the spectral density of the amplitude of the waves from short up to large scales, increasing the distribution of long waves despite the shortwave fluctuations. This dynamics appears to be selfsimilar and possesses a powerlaw behavior in the shortwavelength limit which significantly differs from the exponent obtained via a Kolmogorov dimensional analysis argument. Finally, we show explicitly a tendency to build a longwave coherent structure in finite time.

During, G., Josserand, C., & Rica, S. (2017). Wave turbulence theory of elastic plates. Physica D, 347, 42–73.
Abstract: This article presents the complete study of the longtime evolution of random waves of a vibrating thin elastic plate in the limit of small plate deformation so that modes of oscillations interact weakly. According to the wave turbulence theory a nonlinear wave system evolves in longtime creating a slow redistribution of the spectral energy from one mode to another. We derive step by step, following the method of cumulants expansion and multiscale asymptotic perturbations, the kinetic equation for the second order cumulants as well as the second and fourth order renormalization of the dispersion relation of the waves. We characterize the nonequilibrium evolution to an equilibrium wave spectrum, which happens to be the well known RayleighJeans distribution. Moreover we show the existence of an energy cascade, often called the KolmogorovZakharov spectrum, which happens to be not simply a power law, but a logarithmic correction to the Rayleigh Jeans distribution. We perform numerical simulations confirming these scenarii, namely the equilibrium relaxation for closed systems and the existence of an energy cascade wave spectrum. Both show a good agreement between theoretical predictions and numerics. We show also some other relevant features of vibrating elastic plates, such as the existence of a selfsimilar wave action inverse cascade which happens to blowup in finite time. We discuss the mechanism of the wave breakdown phenomena in elastic plates as well as the limit of strong turbulence which arises as the thickness of the plate vanishes. Finally, we discuss the role of dissipation and the connection with experiments, and the generalization of the wave turbulence theory to elastic shells. (C) 2017 Elsevier B.V. All rights reserved.

During, G., Picozzi, A., & Rica, S. (2009). Breakdown of weakturbulence and nonlinear wave condensation. Physica D, 238(16), 1524–1549.
Abstract: The formation of a largescale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrodinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with smallscale fluctuations superposed (uncondensed particles), which store the information necessary for “time reversal”. We analyze the evolution Of the cumulants of the random wave as originally formulated by DJ. Benney and P.G. Saffman [D.J. Bentley, P.G. Saffman, Proc. Roy. Soc. London A 289 (1966) 301] and A.C. Newell [A.C. Newell, Rev. Geophys. 6 (1968) 1]. This allows us to provide a selfconsistent weakturbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a nonvanishing firstorder cumulant in the hierarchy of the cumulants' equations. More precisely, we show that in the presence of a small condensate amplitude, all relevant statistical information is contained in the offdiagonal second order cumulant, as described by the usual weakturbulence theory. Conversely, in the presence of a highamplitude condensate, the diagonal secondorder cumulants no longer vanish in the long time limit, which signals a breakdown of the weakturbulence theory. However, we show that all asymptotic closure of the hierarchy of the cumulants' equations is still possible provided one considers the Bogoliubov's basis rather than the standard Fourier's (free particle) basis. The nonequilibrium dynamics turns out to be governed by the Bogoliubov's offdiagonal second order cumulant, while the corresponding diagonal cumulants, as well as the higher order cumulants, are shown to vanish asymptotically. The numerical discretization of the NLS equation implicitly introduces an ultraviolet frequency cutoff. The simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters, despite the fact that the theory is expected to breakdown nearby the transition to condensation. The fraction of condensed particles vs energy is characterized by two distinct regimes: For small energies (H << Hc) the Bogoliubov's regime is established, whereas for H less than or similar to Hc the smallamplitude condensate regime is described by the weakturbulence theory. In both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component. The influence of finite size effects and of the dimensionality of the system are also considered. It is shown that, beyond the thermodynamic limit, wave condensation is reestablished in two spatial dimensions, in complete analogy with uniform and ideal 2D Bose gases. (C) 2009 Elsevier B.V. All rights reserved.

Goles, E., & Rica, S. (2011). Irreversibility and spontaneous appearance of coherent behavior in reversible systems. Eur. Phys. J. D, 62(1), 127–137.
Abstract: There is empirical evidence that long time numerical simulations of conservative and reversible partial differential equations evolve, as a general rule (exceptions are the integrable models), towards an equilibrium state that is mainly a coherent structure plus small fluctuations inherent in the conservative and reversible character of the original system. The fluctuations account for the energy difference between the initial configuration and the one of the coherent structure. If the energy is not small enough, then the intrinsic fluctuations may destroy the coherent structure. Thus we arrive to the conclusion that a transition arises from a noncoherent state to a coherent structure as we decrease the initial energy below a critical value. This phenomenon has been successfully observed in various numerical simulations. In this article, we stress that this general behavior is also observed in reversible and conservative cellular automata as in the Q2R model. We point out that this conservative and reversible cellular automata is ab initio deterministic and therefore all our numerical computations are not affected by an approximation of any kind.

Goles, E., Mascareno, A., Medina, P., & Rica, S. (2020). Migrationinduced transition in social structures: a view through the Sakoda model of social interactions. Sci. Rep., 10(1), 18338.
Abstract: We study the dynamics of three populations evolving in a twodimensional discrete grid according to rules of attraction, rejection, or indifference following the framework of the seminal model by Sakoda and we apply it to migration phenomena. An interesting feature of the Sakoda model is the existence of a Pottslike energy which, as a common principle, decreases as time passes by. Here we consider the evolution of two populations until stabilization, then, we perturb this attractor by the inclusion of a third arrival: the immigrants. We show the conditions under which this irruption does not alter significantly the previous attractor (a sociological morphostatic behaviour) or it is dramatically changed (morphogenetic behaviour). We observe empirically that for a morphostatic behaviour the energy decreases while for morphogenesis this energy increases, revealing an escalation of social tension.

Humbert, T., Cadot, O., During, G., Josserand, C., Rica, S., & Touze, C. (2013). Wave turbulence in vibrating plates: The effect of damping. Epl, 102(3), 6 pp.
Abstract: The effect of damping in the wave turbulence regime for thin vibrating plates is studied. An experimental method, allowing measurements of dissipation in the system at all scales, is first introduced. Practical experimental devices for increasing the dissipation are used. The main observable consequence of increasing the damping is a significant modification in the slope of the power spectral density, so that the observed power laws are not in a pure inertial regime. However, the system still displays a turbulent behavior with a cutoff frequency that is determined by the injected power which does not depend on damping. By using the measured damping powerlaw in numerical simulations, similar conclusions are drawn out. Copyright (C) EPLA, 2013

Jarur, M. C., Dumais, J., & Rica, S. (2019). Limiting speed for jumping. C. R. Mec., 347(4), 305–317.
Abstract: General mechanical considerations provide an upper bound for the takeoff velocity of any jumper, animate or inanimate, rigid or soft body, animal or vegetal. The takeoff velocity is driven by the ratio of released energy to body mass. Further, the mean reaction force on a rigid platform during pushoff is inversely proportional to the characteristic size of the jumper. These general considerations are illustrated in the context of Alexander's jumper model, which can be solved exactly and which shows an excellent agreement with the mechanical results. (C) 2019 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
Keywords: Jumping; Takeoff velocity; Locomotion; Limiting speed; Biomechanics

Josserand, C., Pomeau, Y., & Rica, S. (2020). Finitetime localized singularities as a mechanism for turbulent dissipation. Phys. Rev. Fluids, 5(5), 15 pp.
Abstract: The nature of the fluctuations of the dissipation rate in fluid turbulence is still under debate. One reason may be that the observed fluctuations are strong events of dissipation, which reveal the trace of spatiotemporal singularities of the Euler equations, which are the zero viscosity limit of ordinary incompressible fluids. Viscosity regularizes these hypothetical singularities, resulting in a chaotic fluctuating state in which the strong events appear randomly in space and time, making the energy dissipation highly fluctuating. Yet, to date, it is not known if smooth initial conditions of the Euler equations with finite energy do or do not blow up in finite time. We overcome this central difficulty by providing a scenario for singularitymediated turbulence based on the selffocusing nonlinear Schrodinger equation. It avoids the intrinsic difficulty of Euler equations since it is well known that solutions of this NLS equation with smooth initial conditions do blow up in finite time. When adding viscosity, the model shows (i) that dissipation takes place near the singularities only, (ii) that such intense events are random in space and time, (iii) that the mean dissipation rate is almost constant as the viscosity varies, and (iv) the observation of an ObukhovKolmogorov spectrum with a powerlaw dependence together with an intermittent behavior using structure function correlations, in close correspondence with the one measured in fluid turbulence.

Kiwi, M., de Espanes, P. M., Rapaport, I., Rica, S., & Theyssier, G. (2014). Strict Majority Bootstrap Percolation in the rwheel. Inf. Process. Lett., 114(6), 277–281.
Abstract: In the strict Majority Bootstrap Percolation process each passive vertex v becomes active if at least [deg(v)+1/2] of its neighbors are active (and thereafter never changes its state). We address the problem of finding graphs for which a small proportion of initial active vertices is likely to eventually make all vertices active. We study the problem on a ring of n vertices augmented with a “central” vertex u. Each vertex in the ring, besides being connected to u, is connected to its r closest neighbors to the left and to the right. We prove that if vertices are initially active with probability p > 1/4 then, for large values of r, percolation occurs with probability arbitrarily close to I as n > infinity. Also, if p < 1/4, then the probability of percolation is bounded away from 1. (c) 2014 Elsevier B.V. All rights reserved.
Keywords: Bootstrap percolation; Interconnection networks

Mason, P., Josserand, C., & Rica, S. (2012). Activated Nucleation of Vortices in a DipoleBlockaded Supersolid Condensate. Phys. Rev. Lett., 109(4), 5 pp.
Abstract: We investigate theoretically and numerically a model of a supersolid in a dipoleblockaded BoseEinstein condensate. The dependence of the superfluid fraction with an imposed thermal bath and a uniform boost velocity on the condensate is considered. Specifically, we observe a critical velocity for the nucleation of vortices in our system that is strongly linked to a steplike decrease in the superfluid fraction. We are able to use a scaling argument based on the energy required to activate a vortex, relating the critical temperature to the critical velocity, and find that this relationship is in good agreement with the numerical simulations carried out on the nonlocal GrossPitaevskii equation.

Medina, P., Goles, E., Zarama, R., & Rica, S. (2017). SelfOrganized Societies: On the Sakoda Model of Social Interactions. Complexity, , 16 pp.
Abstract: We characterize the behavior and the social structures appearing from a model of general social interaction proposed by Sakoda. The model consists of two interacting populations in a twodimensional periodic lattice with empty sites. It contemplates a set of simple rules that combine attitudes, ranges of interactions, and movement decisions. We analyze the evolution of the 45 different interaction rules via a Pottslike energy function which drives the system irreversibly to an equilibriumor a steady state. We discuss the robustness of the social structures, dynamical behaviors, and the existence of spatial long range order in terms of the social interactions and the equilibrium energy.

Mellado, P., Concha, A., & Rica, S. (2020). Magnetoelectric Effect in Dipolar Clusters. Phys. Rev. Lett., 125(23), 237602.
Abstract: We combine the anisotropy of magnetic interactions and the point symmetry of finite solids in the study of dipolar clusters as new basic units for multiferroics metamaterials. The Hamiltonian of magnetic dipoles with an easy axis at the vertices of polygons and polyhedra, maps exactly into a Hamiltonian with symmetric and antisymmetric exchange couplings. The last one gives rise to a DzyaloshinskiiMoriya contribution responsible for the magnetic modes of the systems and their symmetry groups, which coincide with those of a particle in a crystal field with spinorbit interaction. We find that the clusters carry spin current and that they manifest the magnetoelectric effect. We expect our results to pave the way for the rational design of magnetoelectric devices at room temperature
