un 2 00 8 epl draft Transversal interface dynamics of a front connecting a stripe pat - tern to a uniform state

Interfaces in two-dimensional systems exhibit unexpected complex dynamical behaviors, the dynamics of a border connecting a stripe pattern and a uniform state is studied. Numerical simulations of a prototype isotropic model, the subcritical Swift-Hohenberg equation, show that this interface has transversal spatial periodic structures, zigzag dynamics and complex coarsening process. Close to a spatial bifurcation, an amended amplitude equation and a one-dimensional interface model allow us to characterize the dynamics exhibited by this interface. Introduction. – Non equilibrium processes often lead in nature to pattern formation developing from a homogeneous state through the spontaneous breaking of the symmetries present in the system [1]. In recent decades, much effort has been devoted to the study of pattern formation (see review [2] and the references therein) arising in systems such as chemical reactions, gas discharge systems, CO2 lasers, liquid crystals, hydrodynamic or electroconvective instabilities, and granular matter (see review [3]), to mention a few. A unified description for the dynamics of spatially periodic structures, developed at the onset of a bifurcation, is achieved by means of amplitude equations for the critical modes. Such a description is valid in the case of weak nonlinearities and for a slow spatial and temporal modulation of the base pattern [2]. As an example, the Newell-Whitehead-Segel equation [4] describes the dynamics of a stripe pattern formed in a twodimensional system. Another ubiquitous phenomenon in nature is the interface dynamics or front propagation. The concept of front propagation, emerged in the field of population dynamics [5–7], has gained growing interest in biology, chemistry, physics, and mathematics (See, e.g. [8] and references therein). These interfaces connect two extended states, such as: uniforms states, patterns, oscillatory, standing waves, spatio-temporal chaotic and so forth. In one-dimensional systems, an interface connecting two uniform stable states, the most favorable state—for in(a)E-mail address: [email protected] stance energetically—invades the other one with a constant and unique speed [9, 10]. This speed is zero, that is the front is motionless, at the Maxwell point [11]. The above picture changes, when one considers an interface connecting a pattern state and a uniform one or two patterns. Due to spatial translation symmetry breaking, the interface is motionless in a range of parameters, the pinning range [11–13]. This behavior is called locking phenomenon or pinning effect. In bidimensional dynamical systems, few experimental and theoretical studies have been performed on fronts connecting patterns and uniform states [14–17]. The aim of this letter is to study the dynamical behaviors of a front connecting a stripe pattern to a uniform state. Numerical simulations of a prototype model—the isotropic Swift-Hohenberg equation—show that the locking phenomenon of a flat interface persists. However, this flat interface is nonlinearly transversely unstable, that is, a finite perturbation of this interface leads to the appearance of an initial wave number which is subsequently replaced by zigzag dynamics, which presents a complex coarsening. Increasing the longitudinal interface size, the flat interface exhibits a transversal spatial instability, which originates a periodical structure at the interface. We have termed these interfaces embroideries. In order to explain these behaviors in an unified manner, we make use of the amended Newell-Whitehead-Segel equation and prototype one-dimensional model for the interface, which describe