Stable Static Localized Structures in One Dimension

Abstract

We study the existence, the stability properties, and the bifurcation structure of static localized solutions in one dimension, near the robust existence of stable fronts between homogeneous solutions and periodic patterns.

Localized patterns and hole solutions in one-dimensional extended systems

“…From the dynamic point of view, localized patterns in one-dimensional spatial systems are the homoclinic connection for the stationary dynamical system. Recently, from a geometrical point of view, the existence, stability properties, and bifurcation diagrams of localized patterns in one-dimensional extended systems have been studied [10].…”

“…From the dynamic point of view, localized patterns in one-dimensional spatial systems are the homoclinic connection for the stationary dynamical system. Recently, from a geometrical point of view, the existence, stability properties, and bifurcation diagrams of localized patterns in one-dimensional extended systems have been studied [10].The aim of this manuscript is to describe how one-dimensional localized patterns and hole solutions arise from front interactions. From a prototype model that exhibits localized patterns and hole solutions, the subcritical Swift-Hohenberg equation, we deduce an adequate equation for the envelope of these particle type solutions.…”

“…It is important to remark that the bifurcation diagram shown in Fig. 3 have been deduced from geometrical arguments based in horseshoe behavior of the attractive and repulsive manifold of a ordinary differential equation [10]. In the pining rage, the front solution is motionless.…”

“…Due to the u(x,t) oscillatory nature of the front interaction, which alternates between attractive and repulsive, we can infer the existence, stability properties, and bifurcation diagrams of localized patterns and hole solutions. Hence, we reobtain the bifurcation diagrams of localized patterns and hole solution deduced from horseshoe behavior of the attractive and repulsive manifold of ordinary differential equations [10].…”

Localized States in Bistable Pattern-Forming Systems

“…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].…”