Domain Growth and Topological Defects in Some Nonpotential Problems

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

PhD Thesis (2000)

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.

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.