Domain Growth and Topological Defects in Some Nonpotential Problems

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.