Modulated optical structures over a modulationally stable medium

Extended stable equilibrium invaded by an unstable state

Coexistence of states is an indispensable feature in the observation of domain walls, interfaces, shock waves or fronts in macroscopic systems. The propagation of these nonlinear waves depends on the relative stability of the connected equilibria. In particular, one expects a stable equilibrium to invade an unstable one, such as occur in combustion, in the spread of permanent contagious diseases, or in the freezing of supercooled water. Here, we show that an unstable state generically can invade a locally stable one in the context of the pattern forming systems. The origin of this phenomenon is related to the lower energy unstable state invading the locally stable but higher energy state. Based on a one-dimensional model we reveal the necessary features to observe this phenomenon. This scenario is fulfilled in the case of a first order spatial instability. A photo-isomerization experiment of a dye-dopant nematic liquid crystal, allow us to observe the front propagation from an unstable state.

Multifront regime of a piecewise-linear FitzHugh-Nagumo model with cross diffusion

Oscillatory reaction-diffusion fronts are described analytically in a piecewise-linear approximation of the FitzHugh-Nagumo equations with linear cross-diffusion terms, which correspond to a pursuit-evasion situation. Fundamental dynamical regimes of front propagation into a stable and into an unstable state are studied, and the shape of the waves for both regimes is explored in detail. We find that oscillations in the wave profile may either be negligible due to rapid attenuation or noticeable if the damping is slow or vanishes. In the first case, we find fronts that display a monotonic profile of the kink type, whereas in the second case the oscillations give rise to fronts with wavy tails. Further, the oscillations may be damped with exponential decay or undamped so that a saw-shaped pattern forms. Finally, we observe an unexpected feature in the behavior of both types of the oscillatory waves: the coexistence of several fronts with different profile shapes and propagation speeds for the same parameter values of the model, i.e., a multifront regime of wave propagation.

Spontaneous motion of localized structures induced by parity symmetry breaking 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.

Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems

We propose a route to spatiotemporal chaos for one-dimensional stationary patterns, which is a natural extension of the quasiperiodicity route for low-dimensional chaos to extended systems. This route is studied through a universal model of pattern formation. The model exhibits a scenario where stationary patterns become spatiotemporally chaotic through two successive bifurcations. First, the pattern undergoes a subcritical Andronov-Hopf bifurcation leading to an oscillatory pattern. Subsequently, a secondary bifurcation gives rise to an oscillation with an incommensurable frequency with respect to the former one. This last bifurcation is responsible for the spatiotemporally chaotic behavior. The Lyapunov spectrum enables us to identify the complex behavior observed as spatiotemporal chaos, and also from the larger Lyapunov exponents characterize the above instabilities.

Field patterns in periodically modulated optical parametric amplifiers and oscillators

Spatially localized and periodic field patterns in periodically modulated optical parametric amplifiers and oscillators are studied. In the degenerate case (equal signal and idler beams) we elaborate on the systematic method of construction of the stationary localized modes in the amplifiers, and study their properties and stability. We describe a method of constructing periodic solutions in optical parametric oscillators, by adjusting the form of the external driven field to the given form of either signal or pump beams.

Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems

We derive asymptotically an order parameter equation in the limit where nascent bistability and long-wavelength modulation instabilities coalesce. This equation is a variant of the Swift-Hohenberg equation that generally contains nonvariational terms of the form psinabla(2)psi and /nablapsi/(2). We briefly review some of the properties already derived for this equation and derive it on three examples taken from chemical, biological, and optical contexts. Finally, we derive the equation on a general class of partial differential systems.