Localized States in Bistable Pattern-Forming Systems

Abstract

We present an unifying description close to a spatial bifurcation of localized states, appearing as large amplitude peaks nucleating over a pattern of lower amplitude. Localized states are pinned over a lattice spontaneously generated by the system itself. We show that the phenomenon is generic and requires only the coexistence of two spatially periodic states. At the onset of the spatial bifurcation, a forced amplitude equation is derived for the critical modes, which accounts for the appearance of localized peaks. In one-dimensional systems, localized patterns can be described as homoclinic orbits passing close to a spatially oscillatory state and converging to an homogeneous state [9,10], whereas domains are seen as heteroclinic trajectories joining the fixed points of the corresponding dynamical system [11]. Recently, in a nematic liquid crystal light valve with optical feedback it has been found experimentally a different type of localized states, appearing as a large amplitude peaks nucleating over a lower amplitude pattern and therefore called localized peaks [12]. Similar observations have been reported in a Newtonian fluid when non linear surface waves are parametrically excited with two frequencies [13] and in numerical simulations of an atomic vapor with optical feedback [14]. Recently, longitudinal modes with localized peaks over a spatially modulated background have been shown in numerical simulations of Maxwell-Bloch equations for a semiconductor laser [15].All these different types of localized states appear over a patterned background and thus constitute a different class of structures with respect to the ones appearing over an uniform background. The aim of this manuscript is to show that localized peaks are a generic class of localized states, appearing whenever a pattern forming system exhibits coexistence of two spatially periodic states. The mechanisms that originate this circumstances are more than a few, for instance, one can consider a multi-stable system, which shows two consecutive spatial bifurcations to different states when one parameter is changed. There is a large number of physical systems that display this kind of behavior, therefore there is a vast number of possible models. In order to derive an unifying and simple description of localized peaks, we develop a theoretical model for one-dimensional spatially extended systems close to a spatial bifurcation. The model, which shows coexistence between different patterns and stable front solutions between them, is based on an amplitude equation that includes a spatial parametric forcing. This extension with respect to conventional amplitude equations, allows to describe localized patterns and to account for the main properties of these solutions. The model includes the interaction of the slowly varying envelope with the small scale of the underlying pattern solution [16], well-known as the non-adiabatic effect [17,18].

Localized Turing patterns in nonlinear optical cavities

“…Often, these can be defined as local back and forth switches between two stable states. Hence, they can be of several kinds, depending on the nature of the states involved: both spatially homogenous [1][2][3][4][5], homogeneous and periodic [6,7], both periodic [8], or even periodic and biperiodic in space [9]. The present paper is concerned with the "homogeneous and periodic" case, so that localized states are localized patterns.…”

Localized States in Bi‐Pattern Systems

“…Notice that the Ansatz for (10). Recently, we have derived a phenomenological model in which the spatial forcing has an amplitude proportional to a polynomial development of the slowly varying amplitude [29]. In this case, the nonadiabatic effect is overestimated since the nonresonant terms are proportional to √ μ, however the qualitative dynamics of both models are very similar.…”

“…It is worth to note that in the case of positive γ, the wave number of the pattern is modified by the inverse of the square amplitude R 2 0 , so that patterns with larger amplitude have smaller wave number. At variance, when γ is negative the patterns with larger amplitude have smaller wavelength [29].…”

“…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 [29]. The aim of the present article is to study and characterize the universal mechanism that is at the origin of localized states pinned over an underlying lattice.…”

Optimal liquid crystal modulation controlled by surface alignment and anchoring strength

“…Two standard examples of such devices in widespread use are spatial light modulators [4][5][6][7][8][9][10][11] and photorefractive cells [12][13][14][15][16]. The common mode of operation of these and other liquid crystal devices is that the voltage applied to the liquid crystal has a uniform component, required to move the liquid crystal past the Fredericks transition, and a generally weaker spatially modulated component that creates a nonuniform director alignment field in the liquid crystal.…”