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We present a unifying description of localized states observed in systems with coexistence of two spatially periodic states, called

Spatial patterns appear spontaneously in out-of-equilibrium systems and are observed in many different physical contexts [

In one-dimensional systems, localized structures or localized patterns can be described as homoclinic orbits passing close to a spatially oscillatory state [

A typical experimental profile of a localized peak observed in the LCLV with optical feedback.

The interplay of localized states with a structured background is particularly interesting in nonlinear optics, where it opens interesting technical issues on the control of localized structures, with the possibility of driving their dynamics and tailoring their interactions [

On the basis of amplitude equations, a first preliminary one-dimensional description of localized states observed in bi-pattern systems was done by the present authors in a recent Letter [

As examples of bi-pattern systems in two-dimensions, we consider: a liquid crystal light valve (LCLV) with optical feedback, which provides our experimental framework, and a spatially forced subcritical pitchfork as a prototype model. Both systems show robust existence of localized states, which exhibit different shapes depending on the symmetry of the supporting patterns.

The paper is organized as follows. In Section

To provide inspiration of localized states in bi-pattern systems, we shall consider a liquid crystal valve with optical feedback as an experiment and numerical simulations of a spatially forced subcritical pitchfork as a prototype model. In the experiment, the underlying lattice of the bi-pattern is realized either by letting the systems to spontaneously generate a self-organized pattern, or by imposing an intensity grid through a spatial periodic forcing. This last case compares directly with the model, which is indeed a spatially forced amplitude equation.

The experimental setup, consisting of a LCLV in an optical feedback loop, is the same as the one reported in [

The feedback loop is closed by an optical fiber bundle and is designed in such a way that diffraction and polarization interference are simultaneously present [

As a first set of experiment, we fix

By increasing the input light intensity

Hexagonal patterns and localized peaks observed in the LCLV system for

The theoretical model for the LCLV feedback system was previously derived in [

where

and

the diffraction being accounted for by the operator

We have performed numerical simulations of our model equations (^{2}/mW,

Numerical profile of the intensity showing a localized peak in the LCLV system.

In brief, it exists a large range of parameters within which the LCLV with optical feedback spontaneously is a bi-pattern system, exhibiting localized states pinned over a self-organized lattice generated by the system itself.

A prototype model of bistability is the subcritical pitchfork model

Bifurcation diagram of model (

In order to have a bi-pattern system, we consider the following spatially forced model

Interface between the two hexagonal patterns exhibited by model (

Hence, in the bi-pattern region we observe front solutions and localized states between the two patterns. As a consequence of the interplay between the envelope variations and the wave number of the underlying pattern, the front solutions are motionless in a region of parameters, so-called the pinning range. Close to this region we expect to observe a family of localized states [

Localized peak observed in the spatially forced model (

To illustrate that the localized peaks exhibit different shapes depending on the supporting pattern, we consider the subcritical pitchfork model with an orthogonal forcing

Localized peak observed in the spatially forced model (

Recently, we have shown in the LCLV experiment that a spatially periodic grid can be imposed, by means of a spatial light modulator, on the profile of the input beam [

We have imposed on the system either a hexagonal or a square intensity grid. For both grids, the input intensity is ^{2} with an amplitude modulation of approximately

As examples of bi-patterns, we show in Figures

Interface between the two patterns observed in the spatially forced experiment: (a) square grid, (b) hexagonal grid. The voltage applied to the LCLV is (a) ^{2}.

Localized peak observed in the spatially forced experiment for a square grid. The voltage applied to the LCLV is

We can notice a good qualitative agreement between the experimental profiles and those obtained by numerical simulation of the spatially forced models (

In conclusion, localized states in bi-pattern systems are robust phenomena and appear naturally when bistability is accompanied by a mechanism of spatial periodic forcing. This can be either externally imposed, or generated by the system itself in a certain range of parameters. In the next section we will present an unified description of localized states in one-dimensional bi-pattern systems.

As we have seen, the main ingredient for the appearance of localized peaks is the coexistence of two spatially periodic states, and this, in some sense, regardless the way in which the two patterns have been created. In order to provide a generic description of such a situation, we consider a a one-dimensional spatially extended system that exhibits a sequence of spatial bifurcations as shown in Figure

A typical bifurcation diagram allowing for the appearance of localized peaks: at a certain value of

As depicted in Figure

with

For given values of the parameters, the two stable uniform stationary states of (

In order to take into account the nonadiabatic effect, we compute the corrections of the amplitude equation by including the nonresonant terms, that is, the solvability condition for the amplitude

To illustrate the effect of nonresonant terms we keep the leading term

In order to obtain the change of the front interaction as a result of the spatial forcing, we consider the front solution of the resonant equation

where

in (

with

where

and

As a consequence of the spatial forcing the front interaction law close to the pinning range, (

Oscillatory interaction force between two front solutions. The inset figures are the stable localized patterns observed at the Maxwell points (black dots), where the interaction changes its sign.

Due to the oscillatory nature of the front interaction, which alternates between attractive and repulsive forces (cf. Figure

Bi-pattern systems are spatially extended systems that display coexistence of two pattern states. They exhibit a rich variety of localized solutions, in particular, localized peaks are pinned over a spatial grid, that can be either spontaneously generated by the system itself, or externally imposed. These localized states are of particular interest in nonlinear optics, where they constitute the elementary pixels of optical memories. The interplay with a structured background open new technical perspectives on the control of their dynamics. We have derived an unified and simple description of localized peaks in one-dimensional spatially extended systems close to a spatial bifurcation. This model allows us to understand the mechanism underlying the formation of localized states, which is based on the coupling between the spatial variations of the envelope and the wavelength of the background pattern. In two-dimensions, we show that these localized states persist and exhibit different shapes depending on the symmetry of the supporting patterns. However, a natural extension of the front interaction law, for instance an interface tension, is not yet available. Work in this direction is in progress.

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