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Domain Growth and Topological Defects in Some Nonpotential Problems

Gallego, Rafael (Directors M. San Miguel and R. Toral)
PhD Thesis , (2000)

In chapter 1 we present some important known results about interface dynamics, both in potential and nonpotential systems. Firstly, we present a classification of dynamical systems, including a detailed explanation of the term "nonpotential", frequently used throughout this thesis. The second part of the chapter reviews the basic concepts regarding domain growth and dynamical scaling. Basic results about growth laws for different relevant systems are presented. We also compare several systems in order to determine which are the dominant growth mechanisms. Finally, we introduce the concept of scaling function and explain how it can be used to characterize dynamical scaling. In chapter 2 we consider a model with three coupled fields (Busse-Heikes model [21]), which was proposed to study rotating Rayleigh-B´enard convection. Each field represents the amplitude of a set of parallel convective rolls with a relative orientation of 60Æ with respect to each other. In general, the dynamics is nonpotential and there are three stable phases that coexist, each one associated with one of the three orientations. The rotation angular velocity of the fluid cell is related to nonpotential effects in the model. Above a critical rotation angular velocity, an instability that leads to a cyclic alternation between the modes takes place [K ¨ uppers-Lortz (KL) instability]. In the original version of the model without spatial dependence or noise terms, the system alternates between the three phases. Contrary to what is observed in the experiments, the alternating period diverges with time. We show how this problem can be circumvent with the presence of fluctuations, that are modeled by adding white noise to the equations. Moreover, we give a procedure to calculate the alternating period analytically in a certain range of parameters. In two spatial dimensions, the KL instability is studied by using different kinds of diffusion-like operators. It is observed that operators with anisotropic derivatives lead to an essentially constant intrinsic period of the KL instability, whereas isotropic derivatives lead to the temporal divergence of this period, as happens in the original model without spatial dependence. Outside the unstable KL region, there is a regime in which three competing stable states coexist. In one spatial dimension there is domain growth, and the final state is an homogeneous solution filling up the whole system. We find that this coarsening process is self-similar, with a growth law that possesses two clearly defined dominant behaviors. In two dimensions, the limit of potential dynamics is such that there is domain growth with self-similar evolution. On the contrary, the nonpotential dynamics may inhibit coarsening for large enough system sizes. We study the influence of nonpotential effects on front motion as well as the formation of defects formed by three-armed spirals. These defects, together with the nonpotential dynamics, are responsible for coarsening inhibition in large systems. When only two amplitudes are excited during the growth process, spiral formation is not possible and coarsening takes place. This growth process, as in the case of one dimension, is self-similar, with a growth law different from that of the potential dynamics limit. The study performed in chapter 3 belongs to the general framework of pattern formation in systems with broken symmetries. In particular, we study the effect of a temporal modulation at three times the critical frequency on a Hopf bifurcation. The system is modeled with a complex Ginzburg-Landau equation with an extra quadratic term, resulting from the strong coupling between the external field and unstable modes. The forcing breaks the phase symmetry, and three stable phase locked states appear above a critical forcing intensity. For large forcings, the excitable regime exhibits the same generic properties of the Busse-Heikes model studied in chapter 2. On the other hand we show, both analytically and numerically, the existence of a transition between one-armed phase spirals and three-armed excitable amplitude spirals when the forcing intensity is increased. Driven nonlinear optical systems offer a wealth of opportunities for the study of pattern formation and other nonequilibrium processes in which the spatial coupling is caused by diffraction instead of diffusion. Only very recently domain growth has been considered in some of these systems and some growth laws obtained from numerical simulations have been reported [9, 22, 23, 24]. Nevertheless, clear mechanisms for the growth laws have often not yet been identified. In addition, the question of dynamical scaling has, in general, not been addressed so far. As a clear example of a nonlinear optical system in which the issues of domain growth and dynamical scaling can be addressed and for which detailed clear results can be obtained, we consider in chapter 4 the formation of transversal structures in a optical cavity filled with a nonlinear kerr medium [25, 26]. In a certain range of parameters, when the cavity is illuminated with a linearly polarized input field, domains of stable polarization states emerge. In this situation of optical bistability, we find three different regimes corresponding to the dynamical evolution of such domains, namely, labyrinthine patterns, formation of localized structures and domain coarsening. For the latter we give evidence of the existence of dynamical scaling, with a growth law similar to that resulting from a curvature driven interface motion. Finally, the main conclusions of the thesis are presented in chapter 5.