Front dynamics and pinning-depinning phenomenon in spatially periodic media

A model for micro-front dynamics using a ϕ4 equation

In this article, we propose a numerical model based on the ϕ4 equation to simulate the dynamics of a front inside a microchannel that features an imperfection at a sidewall to different flow rates. The micro-front displays pinning-depinning phenomena without damped oscillations in the aftermath. To model this behavior, we propose a ϕ4 model with a localized external force and a damping coefficient. Numerical simulations with a constant damping coefficient show that the front displays pinning-depinning phenomena showing damped oscillations once the imperfection is overcome. Replacing the constant damping coefficient with a parabolic spatial function, we reproduce accurately the experimental front-defect interaction.

Stationary broken parity states in active matter models

We demonstrate that several nonvariational continuum models commonly used to describe active matter as well as other active systems exhibit nongeneric behavior: each model supports asymmetric but stationary localized states even in the absence of pinning at heterogeneities. Moreover, such states only begin to drift following a drift-transcritical bifurcation as the activity increases. Asymmetric stationary states should only exist in variational systems, i.e., in models with gradient structure. In other words, such states are expected in passive systems, but not in active systems where the gradient structure of the model is broken by activity. We identify a "spurious" gradient dynamics structure of these models that is responsible for this nongeneric behavior, and determine the types of additional terms that render the models generic, i.e., with asymmetric states that appear via drift-pitchfork bifurcations and are generically moving. We provide detailed illustrations of our results using numerical continuation of resting and steadily drifting states in both generic and nongeneric cases.

Steady states and coarsening in one-dimensional driven Allen-Cahn system

We study the steady states and the coarsening dynamics in a one-dimensional driven nonconserved system modeled by the so-called driven Allen-Cahn equation, which is the standard Allen-Cahn equation with an additional driving force. In particular, we derive equations of motion for the phase boundaries in a phase ordering system obeying this equation using a nearest-neighbor interaction approach. Using the equations of motion we explore kink binary and ternary interactions and analyze how the average domain size scale with respect to time. Further, we employ numerical techniques to perform a bifurcation analysis of the one-period stationary solutions of the equation. We then investigate the linear stability of the two-period solutions and thereby identify and study various coarsening modes.

Localized standing waves induced by spatiotemporal forcing

Particle-type solutions are observed in out-of-equilibrium systems. These states can be motionless, oscillatory, or propagative depending on the injection and dissipation of energy. We investigate a family of localized standing waves based on a liquid-crystal light valve with spatiotemporal modulated optical feedback. These states are nonlinear waves in which energy concentrates in a localized and oscillatory manner. The organization of the family of solutions is characterized as a function of the applied voltage. Close to the reorientation transition, an amplitude equation allows us to elucidate the origin of these localized states and establish their bifurcation diagram. Theoretical findings are in qualitative agreement with experimental observations. Our results open the possibility of manipulating localized states induced by light, which can be used to expand and improve the storage and manipulation of information.

Nonlinear Localization of Dissipative Modulation Instability

Modulation instability (MI) in the presence of noise typically leads to an irreversible and complete disintegration of a plane wave background. Here we report on experiments performed in a coherently driven nonlinear optical resonator that demonstrate nonlinear localization of dissipative MI: formation of persisting domains of MI-driven spatiotemporal chaos surrounded by a stable quasi-plane-wave background. The persisting localization ensues from a combination of bistability and complex spatiotemporal nonlinear dynamics that together permit a locally induced domain of MI to be pinned by a shallow modulation on the plane wave background. We further show that the localized domains of spatiotemporal chaos can be individually addressed-turned on and off at will-and we explore their transport behavior as the strength of the pinning is controlled. Our results reveal new fundamental dynamics at the interface of front dynamics and MI, and offer a route for tailored patterns of noiselike bursts of light.

Dissipative Polarization Domain Walls in a Passive Coherently Driven Kerr Resonator

Using a passive, coherently driven nonlinear optical fiber ring resonator, we report the experimental realization of dissipative polarization domain walls. The domain walls arise through a symmetry breaking bifurcation and consist of temporally localized structures where the amplitudes of the two polarization modes of the resonator interchange, segregating domains of orthogonal polarization states. We show that dissipative polarization domain walls can persist in the resonator without changing shape. We also demonstrate on-demand excitation, as well as pinning of domain walls at specific positions for arbitrary long times. Our results could prove useful for the analog simulation of ubiquitous domain-wall related phenomena, and pave the way to an all-optical buffer adapted to the transmission of topological bits.

Front propagation steered by a high-wavenumber modulation: Theory and experiments

Homogeneously driven dynamical systems exhibit multistability. Depending on the initial conditions, fronts present a rich dynamical behavior between equilibria. Qualitatively, this phenomenology is persistent under spatially modulated forcing. However, the understanding of equilibria and front dynamics organization is not fully established. Here, we investigate these phenomena in the high-wavenumber limit. Based on a model that describes the reorientation transition of a liquid crystal light valve with spatially modulated optical forcing and the homogenization method, equilibria and fronts as a function of forcing parameters are studied. The forcing induces patterns coexisting with the uniform state in regions where the system without forcing is monostable. The front dynamics is characterized theoretically and numerically. Experimental results verify these phenomena and the law describing bistability, showing quite good agreement.

Front depinning by deterministic and stochastic fluctuations: A comparison

Driven dissipative many-body systems are described by differential equations for macroscopic variables which include fluctuations that account for ignored microscopic variables. Here, we investigate the effect of deterministic fluctuations, drawn from a system in a state of phase turbulence, on front dynamics. We show that despite these fluctuations a front may remain pinned, in contrast to fronts in systems with Gaussian white noise fluctuations, and explore the pinning-depinning transition. In the deterministic case, this transition is found to be robust but its location in parameter space is complex, generating a fractal-like structure. We describe this transition by deriving an equation for the front position, which takes the form of an overdamped system with a ratchet potential and chaotic forcing; this equation can, in turn, be transformed into a linear parametrically driven oscillator with a chaotically oscillating frequency. The resulting description provides an unambiguous characterization of the pinning-depinning transition in parameter space. A similar calculation for noise-driven front propagation shows that the pinning-depinning transition is washed out.

Front propagation transition induced by diffraction in a liquid crystal light valve

Driven optical systems can exhibit coexistence of equilibrium states. Traveling waves or fronts between different states present complex spatiotemporal dynamics. We investigate the mechanisms that govern the front spread. Based on a liquid crystal light valve experiment with optical feedback, we show that the front propagation does not pursue a minimization of free energy. Depending on the free propagation length in the optical feedback loop, the front speed exhibits a supercritical transition. Theoretically, from first principles, we use a model that takes it into account, characterizing the speed transition from a plateau to a growing regime. The theoretical and experimental results show quite fair agreement.

Splitting in the pinning-depinning transition of fronts in long-delayed bistable systems

We investigate the formation of localized domains through front pinning in a periodically forced, bistable semiconductor laser with long-delayed optoelectronic feedback. At difference with 1D spatially extended systems, the transition from the pinning to the propagation regime occurs via two separated bifurcations, each corresponding to the unpinning of one of the fronts surrounding the localized domain. The bifurcation splitting is systematically explored, unveiling the crucial role played by the forcing frequency. The experimental results are reproduced and interpreted by means of a prototypical model of our system.

Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model

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.

Nonvariational mechanism of front propagation: Theory and experiments

Multistable systems exhibit a rich front dynamics between equilibria. In one-dimensional scalar gradient systems, the spread of the fronts is proportional to the energy difference between equilibria. Fronts spreading proportionally to the energetic difference between equilibria is a characteristic of one-dimensional scalar gradient systems. Based on a simple nonvariational bistable model, we show analytically and numerically that the direction and speed of front propagation is led by nonvariational dynamics. We provide experimental evidence of nonvariational front propagation between different molecular orientations in a quasi-one-dimensional liquid-crystal light valve subjected to optical feedback. Free diffraction length allows us to control the variational or nonvariational nature of this system. Numerical simulations of the phenomenological model have quite good agreement with experimental observations.

Optical wall dynamics induced by coexistence of monostable and bistable spatial regions

When nonequilibrium extended homogeneous systems exhibit multistability, it leads to the presence of domain walls between the existing equilibria. Depending on the stability of the steady states, the dynamics differs. Here, we consider the interface dynamics in the case of a spatially inhomogeneous system, namely, an optical system where the control parameter is spatially Gaussian. Then interfaces connect the monostable and the bistable nonuniform states that are associated with two distinct spatial regions. The coexistence of these two regions of different stability induces relaxation dynamics and the propagation of a wall with a time-dependent speed. We emphasize analytically these two dynamical behaviors using a generic bistable model. Experimentally, an inhomogeneous Gaussian light beam traveling through either a dye-doped liquid crystal cell or a Kerr cavity depicts these behaviors, in agreement with the theoretical predictions.

Asymmetric counterpropagating fronts without flow

Out-of-equilibrium systems exhibit domain walls between different states. These walls, depending on the type of connected states, can display rich spatiotemporal dynamics. In this Rapid Communication, we investigate the asymmetrical counterpropagation of fronts in an in-plane-switching cell filled with a nematic liquid crystal. Experimentally, we characterize the different front shapes and propagation speeds. These fronts present dissimilar elastic deformations that are responsible for their asymmetric speeds. Theoretically, using a phenomenological model, we describe the observed dynamics with fair agreement.

Shaping wave patterns in reaction-diffusion systems

We present a method to control the two-dimensional shape of traveling wave solutions to reaction-diffusion systems, such as, interfaces and excitation pulses. Control signals that realize a pregiven wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation. While the control enforces a desired wave shape perpendicular to the local propagation direction, the wave profile along the propagation direction itself remains almost unaffected. Provided that the one-dimensional wave profile of all state variables and its propagation velocity can be measured experimentally, and the diffusion coefficients of the reacting species are given, the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown.

Photo-isomerization fronts in dye-doped nematic liquid crystals

An experimental study of the photo-isomerization dynamics in dye-doped nematic crystals is reported, which shows that, when the sample is illuminated by a Gaussian beam, and for high enough input power, a transition from the nematic to the isotropic phase takes place in the illuminated area. The two phases are spatially connected via a front propagating outward from the center of the beam and following the local intensity profile and thus inducing a photo-controlled optical aperture. The optical intensity and temperature fields on the sample follow the same dynamical profile. The front dynamics is described by a phenomenological bi-stable model with an inhomogeneous control parameter, directly related to the beam intensity profile.

Front pinning and localized states analogues in long-delayed bistable systems

Localized structures have been observed in many spatially extended systems of either biological, chemical, or physical nature. Here, we study experimentally front pinning and dissipative localized structures in a delayed optical system based on a bistable semiconductor laser with optoelectronic feedback. We observe that many of the concepts known to apply to spatially localized structures also apply in this context, with specificities related to the lack of reversibility symmetry. Numerical simulations based on purely prototypical modeling reproduce very well the experimental findings, which indicates that the results do not depend on the specific physical system under consideration, but are, on the contrary, very generic features of time delayed systems.

Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback

The existence, stability properties, and dynamical evolution of localized spatiotemporal chaos are studied. We provide evidence of spatiotemporal chaotic localized structures in a liquid crystal light valve experiment with optical feedback. The observations are supported by numerical simulations of the Lifshitz model describing the system. This model exhibits coexistence between a uniform state and a spatiotemporal chaotic pattern, which emerge as the necessary ingredients to obtain localized spatiotemporal chaos. In addition, we have derived a simplified model that allows us to unveil the front interaction mechanism at the origin of the localized spatiotemporal chaotic structures.

Pinning-depinning transition of fronts between standing waves

Dynamic behaviors of fronts connecting standing waves, such as the locking phenomenon, pinning-depinning transitions, propagation, and front interactions, are studied. Two systems are considered, a vertically driven pendulum chain and a generalized φ(4) model. Both models exhibit in an appropriate region of parameters bistability between standing waves. In the driven pendulum chain, using a Galerkin expansion we characterize the region of bistability between subharmonic waves for the upright and the upside-down pendulum states. We derive analytically the front dynamics in the generalized φ(4) model, showing regions where fronts are oscillatory or propagative. We also characterize the mechanism of the pinning-depinning transition of fronts between standing waves. Using front interactions we predict the emergence of dissipative localized waves supported on a standing wave and characterize their corresponding homoclinic snaking bifurcation diagrams. All these analytical predictions are confirmed by numerical simulations with quite good agreement.

Front propagation in one-dimensional spatially periodic bistable media

Front propagation in heterogeneous bistable media is studied using the Schlögl model as a representative example. Spatially periodic modulations in the parameters of the bistable kinetics are taken into account perturbatively. Depending on the ratio L/l (L is the spatial period of the heterogeneity, l is the front width), appropriate singular perturbation techniques are applied to derive an ordinary differential equation for the position of the front in the presence of the heterogeneities. From this equation, the dependence of the average propagation speed on L/l as well as on the modulation amplitude is calculated. The analytical results obtained predict velocity overshoot, different cases of propagation failure, and the propagation speed for very large spatial periods in quantitative agreement with the results of direct numerical simulations of the underlying reaction-diffusion equation.

Origin of the pinning of drifting monostable patterns

Under drift forces, a monostable pattern propagates. However, examples of nonpropagative dynamics have been observed. We show that the origin of this pinning effect comes from the coupling between the slow scale of the envelope to the fast scale of the modulation of the underlying pattern. We evidence that this effect stems from spatial inhomogeneities in the system. Experiments and numerics on drifting pattern-forming systems subjected to inhomogeneous spatial pumping or boundary conditions confirm this origin of pinning dynamics.

Effects of translational coupling on dissipative localized states

Nonequilibrium localized states under the influence of translational coupling are studied experimentally and theoretically. We show that localized structures are deformed and advected in the direction of the coupling, thus undergoing different instabilities. Experimentally, localized structures are obtained in a light valve with optical feedback. By introducing a tilt of one mirror in the feedback loop, localized structures acquire a translational coupling. To understand the phenomenon in a universal framework we consider a prototypical model of localized states with translational coupling in one and two spatial dimensions. The model allows us to analytically characterize the propagation speed and the deformation exhibited by the localized state profiles as well as to figure out different mechanisms of destabilization of these dissipative structures. The results are in good qualitative agreement with the experimental and numerical observations.

Symmetry-induced pinning-depinning transition of a subharmonic wave pattern

The stationary to drifting transition of a subharmonic wave pattern is studied in the presence of inhomogeneities and drift forces as the pattern wavelength is comparable with the system size. We consider a pinning-depinning transition of stationary subharmonic waves in a tilted quasi-one-dimensional fluidized shallow granular bed driven by a periodic air flow in a small cell. The transition is mediated by the competition of the inherent periodicity of the subharmonic pattern, the asymmetry of the system, and the finite size of the cell. Measurements of the mean phase velocity of the subharmonic pattern are in good agreement with those inferred from an amplitude equation, which takes into account asymmetry and finite-size effects of the system, emphasizing the main ingredients and mechanism of the transition.

Homoclinic snaking of localized patterns in a spatially forced system

Dissipative localized structures exhibit intricate bifurcation diagrams. An adequate theory has been developed in one space dimension; however, discrepancies arise with the experiments. Based on an optical feedback with spatially modulated input beam, we set up a 1D forced configuration in a nematic liquid crystal layer. We characterize experimentally and theoretically the homoclinic snaking diagram of localized patterns, providing a reconciliation between theory and experiments.