The dynamics of patterns

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THE D\TNAMilCS OF

KlflDHB

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®

THE DYNAMICS OF

PATTERNS MI Rabinovich University of California, San Diego

A B Ezersky Russian Academy of Sciences

P D Weidman University of Colorado

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

THE DYNAMICS OF PATTERNS Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4055-4 ISBN 981-02-4056-2 (pbk)

Printed in Singapore by FuIsland Offset Printing

To our parents,

Dora and Israel ovetlana ana Doris Ora ana Merle

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Preface

The creative process of writing this book was less a result of that which arises out of ones own head, hand, pen and paper than a result of the in­ teractive dialogue between authors with a similar vision. Our desire is to present a clear and concise presentation of principal phenomena, basic con­ cepts, and novel ideas in pattern formation using the language of nonlinear dynamics supplemented by photographs of actual patterns observed in the laboratory. We believe, perhaps a little presumptuously, that the principle we adhere to, namely "one chapter - one topic" together with a careful se­ lection of examples, has enabled us to write a book that is comprehensible to graduates, postgraduates, and those researchers who have just started investigating this extremely fascinating field of science. As is well known, the elegance of simplicity tempts one to simplify real phenomena by fitting them to conventional and universal models. We hope that we succeeded in avoiding this temptation unduly, since every chapter is focused on specific experiments. The text is supplemented with two appendices. Appendix A — A Short Guide to Nonlinear Dynamics — is provided as a review of the theoretical underpinnings of nonlinear systems and Appendix B — Key Experiments in Pattern Formation — is included for readers curious about experimental setups designed to observe pattern phenomena in the laboratory. For the most part, the book is based on the lectures read by MIR to students of the University of Nizhny Novgorod, the University of Chicago, and the University of California at San Diego. Some sections are the result of laboratory experiments spearheaded by ABE and PDW. We greatly appreciate numerous discussions on the topics presented vii

Vlll

Preface

herein with H. Abarbanel, V. Afraimovich, I. Aranson, A. Gaponov-Grekhov, L. Kadanov, J. Lebowitz, A. Libchaber, Ya. Sinai, L. Tsimring and P. Varona. The authors are indebted to F. Hjguera for a careful reading of Chapter 10 and to J. Meiss and B. Polotovsky for critical comments concering Appendix A. In addition, L. Tsimring wrote Chapter 11 upon request of the authors. We are ever indebted to Nadezhda Krivatkina and Leonid Rubchinsky who helped us prepare the manuscript, to Michael Sprague for suggestions to improve the layout, and to Kendall Hunter for rescuing files sent to Boulder from Nizhny Novgord. MIR acknowledges support from the En­ gineering Research Program of the Office of Basic Energy Science at the Department of Energy and personally Robert Price who is deeply interested in the subjects discussed in this book. We also appreciate the support of International Center for Advanced Studies in Nizhny Novgorod.

M. I. Rabinovich A. B. Ezersky P. D. Weidman Boulder, Colorado August 2000

Contents

Preface

vii

Chapter 1

Patterns: Prelude to a Dynamical Description

Chapter 2

Linear Stage of Pattern Formation

1 15

Chapter 3 Model Equations 3.1 Swift-Hohenberg equation 3.2 Newell-Whitehead-Segel equation 3.3 Coupled amplitude equations 3.4 Phase equations

27 29 35 38 42

Chapter 4 The Ginzburg-Landau Equation 4.1 The dissipative Ginzburg-Landau equation 4.2 Nerve membrane excitation and the CGL equation 4.3 Optical dynamics and the CGL equation 4.4 Simple patterns in the CGL equation 4.5 Phase equations revisited 4.6 Gallery of phenomena

45 46 48 50 52 55 57

Chapter 5

63

'Crystal' Formation

Chapter 6 Quasicrystals 6.1 Octagons, decagons, and dodecagons 6.2 A generalized Swift-Hohenberg model 6.3 The 'turbulent' crystal ix

75 78 81 82

x

Contents

Chapter 7 Breaking of Order 7.1 A simple model for domain walls 7.2 Topological defects 7.3 The birth of penta-hepta defects 7.4 Dislocations and domain walls in Faraday ripples

87 89 92 96 102

Chapter 8 Localized Patterns 8.1 Bistable media 8.2 Dynamical disorder of structures 8.3 Particle interaction 8.4 Chaotic scattering

107 107 114 115 118

Chapter 9 Spirals 9.1 Active spirals 9.1.1 Spirals in 9.1.2 Spirals in 9.2 Passive spirals 9.2.1 Spirals in 9.2.2 Spirals in

129 133 the complex Ginzburg-Landau equation . . . 133 the FitzHugh-Nagumo model 136 142 the Faraday experiment 143 Rayleigh-Benard convection 145

Chapter 10 Patterns in Oscillating Soap Films 10.1 Introduction 10.2 Observations 10.3 Models for vorticity generation 10.3.1 Marangoni wave model 10.3.2 The role of air

151 151 152 158 160 162

Chapter 11 Patterns in Colonies of Microorganisms 11.1 Dictyostelium discoideum 11.2 Esherichia coli 11.3 Bacillus subtilis

173 174 178 184

Chapter 12 Spatial Disorder 12.1 Introductory remarks 12.2 Characteristics of space series 12.3 The Grassberger-Procaccia algorithm 12.4 Qualitative description of developing disorder 12.5 Dynamical dimension of defect-mediated turbulence

189 189 193 197 199 202

Contents

xi

Chapter 13 Patterns in Chaotic Media 13.1 Introductory remarks 13.2 Chaotic synchronization 13.3 Coexistence of regular patterns and chaos 13.4 Coarse grain spatio-temporal patterns 13.5 Coherent patterns on a chaotic checkerboard

205 205 208 211 214 220

Chapter 14 Epilogue: Living matter and dynamic forms 14.1 Hallucinations 14.2 Spatio-temporal patterns and information processing

225 227 234

Appendix A A Short Guide to Nonlinear Dynamics A.l Dynamical systems A.1.1 Types of dynamical systems A.1.2 Equilibrium states A. 1.3 Homoclinic and heteroclinic trajectories A.1.4 Limit cycles A. 1.5 Quasiperiodic motion A.2 Bifurcations A.2.1 Bifurcations of equilibrium states A.2.2 Birth of periodic motions A.2.3 Change of stability in periodic motions A.3 Chaotic Oscillations A.3.1 Characteristics of chaos and the strange attractor A.3.2 Chaotic Hamiltonian systems A.3.3 Chaotic self-excited oscillations A.4 Synchronization of oscillations A.5 Dynamical chaos and turbulence

239 239 240 241 245 248 249 250 252 258 260 261 261 263 264 266 271

Appendix B Key Experiments in Pattern Formation B.l Parametrically excited patterns B.l.l Experiments with liquids B.1.2 Experiments with granular material B.2 Thermal convection B.2.1 Rayleigh-Benard convection B.2.2 Patterns in Rayleigh-Benard convection B.2.3 Benard-Marangoni convection B.3 Diffusive chemical reactions

...

279 279 279 287 292 292 295 297 302

xii

Contents

B.3.1 Turing patterns

302

B.3.2

306

Oscillating chemical reactions

Bibliography

309

Index

320

Chapter 1

Patterns: Prelude to a Dynamical Description

The theory has to be as simple as possible but not simpler. Albert Einstein The diversity of patterns existing in the surrounding world usually arouses amazement not only in fiction novels but in scientific literature also. Perhaps still more curious is the fact that these diversified patterns are universal to a certain extent: hexagonal patterns on a giraffe's skin and hexagonal structures emerging at the onset of convection in a liquid layer; isolated patterns in colonies of microorganisms and soliton structures in hydrodynamics and in nonlinear optics; spiral chaos in heart muscle fibril­ lation and spiral turbulence in chemical systems; and a multitude of other examples. What is the nature of this universality? Why do analogous spatial pat­ terns arise in systems that are so different? Answers to these questions can be found using key results gleaned from the classical theory of oscil­ lations. It suffices here to recall that the type of oscillations — periodic, quasiperiodic, chaotic, harmonic, or relaxational — is determined by the dynamical properties of the system (e.g. the number of degrees of freedom, the type of nonlinearity, and the presence of fast and slow motions) inde­ pendent of whether the system is acoustical, optical, hydrodynamical, or chemical. An analogous situation exists for the birth and development of spatial patterns. It might seem at first sight that an infinite diversity of such patterns makes their unified dynamical description a hopeless, even meaningless task. However, here again one can exploit the knowledge ac­ cumulated in the classical theory of oscillations, in particular the ideas put forth by Andronov in 1931 concerning the structural stability of dynamical l

2

Patterns:

Prelude to a Dynamical

Description

systems and the investigation of their bifurcations. We briefly review the es­ sential points of these ideas, the roots of which are traced back to Poincare. In his book La Valeur de la Science in the chapter entitled Analysis and Physics, Poincare (1905) wrote that "The main thing for us to do with the equations of mathematical physics is to investigate what may and should be changed in them." Indeed, any description of real life is a model, and in modeling the dynamics of nonlinear systems one is usually confronted with ordinary or partial differential equations containing different nonlinear dependencies. Ideally it would be great to obtain their general solutions and thus predict the behavior of a given model subject to specific initial conditions. However, it is only by rare luck that one is able to find all integrals of a nonlinear system of differential equations. Andronov's remarkable approach toward understanding such systems contained three key points: (i) Only those models demonstrating motions which do not vary with small changes of the parameters can be regarded as really interesting ones — Andronov referred to these as models or dy­ namical systems that arc structurally stable; (ii) To obtain insight into the dynamics of a system means to clarify all principal types of its behavior under all possible initial conditions, i.e. one should investigate the behav­ ior of the model as a whole rather than find particular solutions under specific initial conditions — hence Andronov's fondness of the methods of phase space analysis; and, finally, (iii) Consideration of the behavior of the system as a whole allows one to introduce the concept of topological equiv­ alency of dynamical systems, and requires an understanding of local and global bifurcations as control parameters are varied. Conservation of the topology of the phase portrait corresponds to a qualitatively stable motion of the system with small variation of governing parameters. Partitioning of the space of parameters into regions with different behavior then furnishes a complete picture of the potentialities of a dynamical model (Andronov et ai, 1973a, 1973b). Under certain idealizations it is natural to consider the process of pat­ tern formation as the life of a dynamical system. The features of this life as t —> oo then determine the characteristic aspects of the spatial patterns, be they perfect or irregular. Let us consider the experimental results reproduced here in Figures. 1.1, 1.2 and 1.3 taken from three different investigations of patterns formed on the surface of a liquid. Figure 1.1 is a photograph of a regularly rough free surface of progressive water waves a channel; Fig. 1.2 exhibits a hexagonal

Patterns:

Prelude to a Dynamical

Description

3

Fig. 1.1 Spilling breaking waves on a water surface; the waves propagate from left to right. Taken from (van Dyke, 1982).

lattice of Rayleigh-Benard convection (see Appendix B.2); and Fig. 1.3 displays an irregular pattern of capillary ripples formed on a liquid layer contained in a vertically vibrating cavity (this so-called Faraday experiment is discussed in Appendix B.l). In viewing these patterns many questions arise: Why are the patterns steady? How are they formed out of arbitrary initial conditions? In what way does the pattern change with the variation of experimental parameters? How is one able to describe the patterns? The latter question seems to be simpler than the others and we therefore take it as a starting point in the construction of a dynamical theory of space patterns. The first observation that catches the eye is the existence of a definite spatial scale. It is the characteristic length describing the spanwise and streamwise extent of the propagating wave system in Fig. 1.1 and in Fig. 1.2 it is the well-defined hexagonal scale of instability. Thus it seems natural to suppose that the observed spatial field can be represented as a su­ perposition of plane waves having wavenumber vectors of equal magnitude, viz. u(r,0 = 5Ian(«)eik"r+c.c.

(1-1)

n

where the coefficients an(t) are scalar quantities and all wave vectors k n satisfy |k n | = ko. Then, four modes having wave vectors at right angles to one another as in Fig. 1.4a correspond to a lattice with square cells, and six plane waves with wave vectors forming an angle of 60° with one another as

4

Patterns:

Prelude to a Dynamical

Description

Fig. 1.2 Shadowgraph image of hexagonal cells obtained in Rayleigh-Benard convection in a thin horizontal layer of CO2 under pressure. Taken from (Bodenschatz et al., 1991).

in Fig. 1.4b correspond to a lattice of hexagonal cells. The capillary pattern shown in Fig. 1.3 may be formed by superposing a sufficient number of spatial modes with wave vectors of equal magnitude, provided they are randomly distributed in angle as in Fig. 1.4c. The com­ puter image of a simple random mode distribution in Fig. 1.5 goes a long way toward mimicing the actual experimental snapshot of Fig. 1.3. There exist, between the limiting purely periodic and irregular field distributions in space, numerous intermediate patterns having different de­ grees of irregularity (or order). One of the most remarkable achievements obtained in recent years is the discovery of quasicrystal order in hydrodynamic flows which appears, for example, in capillary ripples parametrically excited by an oscillating gravitational field (Christiansen et al., 1992; Ed­ wards and Fauve, 1993); see Chapter 6 for theoretical details. Typical quasicrystal patterns are presented in Figs. 1.6 and 1.7. Computer simulations readily reproduce these quasiperiodic patterns for which long-range order is typical by incorporating eight (Fig. 1.8a) or twelve (Fig. 1.8b) plane waves of equal wave vector magnitude forming, respectively, angles of 45° and 30° with one another. However, the fact

Patterns:

Fig. 1.3 1986).

Prelude to a Dynamical

Description

5

Chaotic pattern of free surface capillary ripples. Taken from (Rzersky et al.,

K

*>• ***

,y-~~

V Ky'"

/S 5?\. ^]'~ \y " --..

a

b

v

c

Fig. 1.4 Wavenumber vectors |k|„ = ko of spatial modes forming (a) a square pattern, (b) a hexagonal pattern, and (c) a disordered pattern.

that a quasicrystal may be constructed using only eight (or twelve, or n) modes does not explain the existence of the patterns observed. Indeed, why are eight (or twelve) modes persistently encountered? What, generally, determines their number? Is it possible to change the type of quasicrystal symmetry by changing the parameters as, for instance, the frequency of an oscillatory gravitational field or the depth of the fluid layer? Here, again, a dynamical theory is needed to resolve these questions. Figure 1.9 shows the evolution of Benard-Marangoni convection from

6

Patterns:

Prelude to a Dynamical

Description

Fig. 1.5 Computer simulation of a disordered pattern with a spatial spectrum like that in Fig. 1.4c. Taken from (Blumel et al., 1992).

Fig. 1.6 Eight-fold quasicrystal pattern on a liquid layer driven by harmonic vertical forcing. The image of ripples is formed by light reflected from the free surface. Taken from (Christiansen el al., 1992).

a random initial state in a laboratory experiment. It is clear that the field becomes more ordered with time. For this situation the principal questions to ask of the dynamical theory are: What kind of structure will be established as t -> oo? Will it be a regular hexagonal lattice or will it have some disorder in the form of defects? If the latter, how will the defects evolve in time? Of what must a dynamical theory of patterns consist? The principal ob­ jectives of such a theory are: (i) an analysis of the hierarchy of instabilities and the birth of various patterns in the course of evolution from an ini­ tial state, (ii) an investigation of the mutual transformation of patterns as control parameters are varied, (iii) an analysis of possible hysteretic effects

Patterns:

Prelude to a Dynamical

Description

7

Fig. 1.7 Twelve-fold quasicrystal pattern on a liquid layer driven by biharmonic external forcing; experiments in (a) a circular cell and (b) a cell with boundary in the shape of France. Taken from (Edwards and Fauve, 1993).

Fig. 1.8 Wavenumber vectors of spatial modes forming (a) eight-fold and (b)twelve-fold quasicrystals.

in the system, and (iv) an investigation of differences between hand-made spatial chaos (see, for example, the painting by Pollack in Fig. 1.10) and dy­ namical chaos. Much can be done in this direction employing a traditional mode description of nonlinear fields, i.e. replacing the governing PDE's by their finite-dimensional ODE models. In this manner dynamical systems appear in a very natural fashion: for example, the time-dependent mode amplitudes an(t) in Eq. (1.1) may be described by ODE's. Many results of the classical theory of dynamical systems can be used directly if, and when, an appropriate physical meaning may be attached to the variables in the

8

Patterns: Prelude to a Dynamical Description

Fig. 1.9 Snapshots of B6nard-Marangoni convection at three successive times: (a) t = to, (b) t = t\ > to and (c) t = ti >t\. Taken from (Gaponov-Grekhov and Rabinovich, 1990).

Fig. 1.10 Oil on canvas painting entitled "Lavender Mist: Number 1" by Jackson Pol­ lock (1950); National Gallery of Art, Washington, DC.

model equations. Another important aspect of the theory to be developed is associated with the notion of a spatial dynamical system where one (or more) spatial coordinates play the role of time. A simple example is given by RayleighBenard convection which stably manifests ordered patterns that contain domains of rolls having different orientations. Such a pattern is depicted in Fig. 1.11. These types of problems will be considered in Chapter 12.

Patterns:

Prelude to a Dynamical Description

9

Fig. 1.11 Domain rolls observed in Rayleigh-B^nard convection in a circular cavity. Taken from (Ahlers et al., 1985).

Fig. 1.12

Domain wall numerically computed from Eqs. (1.2).

Here we just note that in special cases where a two-dimensional system can be transformed into a one-dimensional problem with spatial variable x, domains of uniform rolls are associated, as x -> +00 and x —► —00, with different equilibrium states of the spatial dynamical system. Moreover, an isolated trajectory (separatrix) connecting these equilibrium states cor­ responds to a boundary between domains, a so-called domain wall. This one-dimensional spatial dynamical system can be described by the following

10

Patterns:

Prelude to a Dynamical

Description

Fig. 1.13 A quasistable bound state of dislocations: a domain wall in parametrically excited ripples. Circles mark dislocations that propagate down to form the inclined domain wall. Taken from (Ezersky et ai, 1995).

pair of amplitude equations

^

= A.-lAl + pADA,

££■

=

A2-[Al + pAl]A2

(1.2)

in which A\(x) and A2(x) are the amplitudes of the modes forming the pattern and p > 1 is the coefficient of mode competition (see Chapter 7 for details). Separatrices of the dynamical system (1.2) meeting the condition Ai(+oo) = 1, A2(+oo) = 0 and A\(—00) = 0, A2{—00) = 1 correspond to a jump in mode amplitude across the domain wall. The pattern described by Eqs. (1.2) exhibits a domain wall like that displayed in Fig. 1.12. When the roll axes are slightly inclined with one another, a pattern like that in Fig. 1.13 is formed (Ezersky et ai, 1995). Circles in this figure mark individual dislocations that evolve in a chain-like fashion to form the inclined domain wall observed in the figure. If several different modes coexist, their nonlinear interaction is likely

Patterns:

Prelude to a Dynamical

Description

11

Fig. 1.14 Pattern in Rayleigh-Benard convection in a layer of CO2 under pressure at a Prandtl number near one. Taken from (Morris et al., 1993).

to produce an irregular or chaotic solution of a spatial dynamical system. This is the simplest example of deterministic disorder to which an invari­ ant chaotic set corresponds in phase space. Field distributions (of velocity, temperature, concentration, etc.) are called order parameters. In the next level of disorder, singularities of the order parameters can exist giving rise to more complicated dynamical patterns such as that shown in Fig. 1.14 for Rayleigh-Benard convection at Prandtl numbers near unity. This is an example of the recently discovered spatial disorder known as spiral-defect chaos; see, for example, (Morris et al., 1993). It consists of a disordered pattern of convection rolls featuring the spontaneous appearance and dis­ appearance of rotating spiral defects. One feature of spiral defects that differentiates them from defects in many other systems is that they are not constrained to be created in pairs. The spiral-defect state behaves as an individual element comprised of single spirals with clockwise or counter­ clockwise winding. The mechanics for the creation of this variety of forms is not yet completely understood, but numerical simulations show that a crucial element for the formation of such patterns and individual spirals is the strength of the mean-drift field. Further and more detailed discussion of spiral defects will be given in Chapter 9. How may disordered patterns be understood? When discussing the con­ cept of disorder even physicists have quite different associations. Experts

12

Patterns:

Prelude to a Dynamical

Description

in statistical physics think of the disordered arrangement of molecules in a gas or liquid at an arbitrary moment of time, whilst solid-state physicists envisage the disorder of magnetic domains and spin orientation. Gener­ ally speaking, disorder is a complex, irregular spatial distribution of some elements or fields. Traditional characteristics of disorder are closely con­ nected with the determination of correlation lengths. For instance, when concerned with long-range order we have in mind a situation where the dis­ tribution of a certain physical quantity (order parameter) in a given region of space is unambiguously related to its value at an infinitely distant region. Perhaps the most familiar example is long-range order in crystalline mate­ rials where a density correlation exists. A well known case for which only short-range order is possible is in the structure liquids; here the random motion of atoms gives rise to density fluctuations that destroy any possible long-range order. However, it has been relatively recently discovered that long-range crystal order does exist in hydrodynamics, a branch of physics in which crystal-like structures are not traditionally contemplated. This topic will be discussed in Chapters 5 and 6. The dynamical theory of disorder describes irregular spatial field distri­ butions, such as the electron density distribution in crystals or the density distribution of matter in galaxies, employing methods of nonlinear dynam­ ics. We are trying to understand whether spatially irregular distributions of physically meaningful fields may have a dynamical origin. In other words, might there exist irregular field distributions that can be described by dy­ namical models? It is anticipated that if the fields have physical origin, they will be described by partial differential equations or some more spe­ cific equations for dynamical systems with several 'times,' where the spatial coordinates represent the 'times.' Even the simplest spatio-temporal analogy convinces us that finitedimensional disorder — an irregular spatial field distribution described by a dynamical system possessing a finite number of degrees of freedom — must exist in Nature and must be no less prevalent than finite-dimensional tem­ poral chaos. There are plenty of examples such as stationary chaotic waves observed in different physical systems. There arises a natural impulse to apply results of the theory of temporal dynamical chaos to problems involv­ ing spatial dynamical disorder. The application is trivial to a certain extent if we have one spatial coordinate and a steady regime. However, very diffi­ cult problems arise even for systems described by a single spatial coordinate. The main question is: Does spatially disordered initial data in that system

Patterns:

Prelude to a Dynamical

Description

13

evolve deterministically in time? When we consider two-dimensional (and especially three-dimensional) spatial disorder of a dynamical origin, quali­ tatively new problems arise. The commensurability-quasiperiodicity-chaos transition (see Appendix A.5) is well known in simple dynamical systems involving low-dimensional ordinary differential equations. By replacing the time variable with a single space coordinate, we can expect to find the same transition scenario for spatial disorder in one-dimensional media. Is such a sequence possible through the variation of a governing parameter for a twoor three-dimensional field? What kind of dynamical system will describe such disorder? We have no definite answer to these questions yet. However, the classical commensurability-quasiperiodicity-chaos transition scenario is highly probable. Returning to the mode superposition description of disorder, it is per­ haps not so surprising that a randomly disordered field like the one shown in Fig. 1.3 may be described by the superposition of a large number of harmonic waves of equal wavelengths. Indeed, what else could be expected if the mode amplitudes are random, their phases are arbitrary, and the wave vector orientations are also random as in Fig. 1.4c? Note that space correlations of a Gaussian amplitude distribution u(r) decay rather fast in accord with the following formula found by (Berry, 1983) K(X) = Ju(T)u(r

+ X)d2r

~ J0(k0\x\)

(1-3)

in which Jo is the Bessel function of order zero, k0 is the wave vector modulus, and x ls the argument of the correlation function. The close similarity between the computer imaging of random mode superposition in Fig. 1.5 with the snapshot of the distribution of capillary waves in Fig. 1.3 produces a very important question: Why have the wave amplitudes estab­ lished a Gaussian distribution with arbitrary phases not synchronized in time? These facts must be accounted for by any realistic dynamical theory.

Chapter 2

Linear Stage of Pattern Formation

Prediction is difficult, especially if it concerns the future. An old Chinese proverb How does an initial, absolutely symmetric (homogeneous) state of a medium or field become inhomogeneous? This usually results from a spon­ taneous breaking of symmetry which represents the change of stability of a stationary state. Symmetry breakings (c/. Appendix A.2) are frequently observed, sometimes without realizing it. Consider making coffee directly from a uniform horizontal layer of granulated coffee beans floating on the surface of water in a broad saucepan. What will be observed? The layer of steam that facilitates extraction of the coffee rises from the bottom of the vessel and disintegrates into individual bubbles thereby producing a coffee surface in the form of hills and valleys. If we remove the saucepan from the stove before it comes to a boil, the coffee particles that have slipped down to the valleys will 'remember' this inhomogeneous state which corre­ sponds to a spontaneous breaking of spatial symmetry. Such a transition occurs monotonically in time and is often referred to as a Turing bifurca­ tion (Turing, 1952) or an exchange of stability (Davis, 1969), unlike the Andronov-Hopf bifurcation which corresponds to a transition to an oscilla­ tory instability (see Appendix A). Turing revealed this exchange of stability in chemical reactions with diffusion nearly a half century ago and was the first to establish a connection between the spontaneous birth of inhomoge­ neous patterns with the mechanism of morphogenesis in biology. See, for instance, the nice review by (Koch and Meinhardt, 1994) where one can find many interesting examples of Turing patterns in biology. Turing showed that under certain conditions even just two interacting chemicals can gen15

16

Linear Stage of Pattern Formation

erate stable inhomogeneous nonoscillatory patterns if one of the substances diffuses much faster than the other. This result is not at all intuitive since diffusion is expected to smooth out concentration differences rather than produce them. The exchange of stability bifurcation may be investigated using the diffusion reaction equations

|=f(u,«+DV2u

(2.1)

where u is the population vector, f (u, /?) determines the point reaction ki­ netics with control parameter /3, and D is the diffusion matrix. A standard linear stability analysis for a uniform density distribution, u = uo, with respect to spatial perturbations of wavenumber k in the form e'(k'""', gives a complex dispersion relation from which the Lyapunov exponent A(k,/3) and frequency w(k, /?) are determined. The critical wavenumber k c corre­ sponding to the minimum value of the control parameter pc at which the growth rate a = Re(A) first becomes positive may also be determined from this dispersion equation. The characteristic feature of the Turing instability corresponding to the class of space-symmetry-breaking instabilities is that A is real, i.e. u = Im(A) = 0; thus there are no temporal oscillations at the onset of instability. Owing to the symmetry properties of Eq. (2.1), which represent the properties of homogeneity and isotropy of the solution in which the chemical reaction occurs, the instability rate a depends only on the magnitude fco of wavenumber k and not on its direction. In other words, the linear stage of spontaneous symmetry breaking simply increases the field strength which, in general, may be represented in the form /•2JT

[A^e^-T

u(r, t) = uo+

+ A^e~ik*-r] dip

(2.2)

Jo in which Av and kv are the amplitude and wavenumber distributions for waves propagating in the direction 8 = tp and |ky,| = ko. Of course, the kind of spatial pattern realized as t —¥ oo may be determined only by application of some nonlinear theory. A simple example that demonstrates the onset of an inhomogeneous density distribution in the absence of temporal oscillations is the Brussela-

Linear Stage of Pattern

Formation

17

tor model (GlansdorfF and Prigogine, 1971) — = a + u2v - (1 + 0)u + at

DuV2u

-^-=0u-u2v + DvV2u. (2.3) at These equations describe the spatio-temporal dynamics of the intermediate components u and v in an autocatalytic chemical reaction with respective diffusion coefficients £>u and Dv. The reactions a —y u 2u + v -^> 3u

u-^c

(2.4)

describe the concentration of the original substances a and /3 for which the final products c and d are constant when all reaction rates 9?j equal unity. Setting the left-hand sides of Eqs. (2.3) to zero gives the uniform steadystate solution: u 0 = a, vo = 0/a- Inserting then u = a + u and v = - +v in Eqs. (2.3), and linearizing for small perturbations u and v, yields the pair of disturbance equations d_ dt where the linearized operator L is . L =

( P - 1 + DUV2 ( -0

-a

2

a2 \ + DvV2)-

(2 6)

-

Stability of the trivial solution u = v = 0 of Eq. (2.5) is determined by solv­ ing for the eigenvalues A„ corresponding to the eigenfunctions (un,vn) = (U„, V„)e*k"'r of L, since arbitrary perturbations can be written in the form

($)-E^'(t)If for some k„ one finds Re(A„) > 0 then the steady-state solution is unstable. Suppose for simplicity that u and v depend on the single space

Linear Stage of Pattern

Formation

Im A.

ImX

Re A

Fig. 2.1 Qualitative features of the eigenvalue A of Eq. (2.8) in (p,q) parameter space; the parabolic curve is q = ^ p 2 .

coordinate x in which case (iii,i)i) = (U\, Vi)elkx. Eq. (2.5) yields the characteristic equation for A

Then substitution into

A2 + pX + q = 0

(2.8)

where the coefficients p and q are given by p=a2

+ l-0

+ (Du + Dv)k2

q = a20 - (/? - 1 - k2Du)(a2

+

k2Dv).

(2.9)

The variation of Re(A) and Im(A) in (p, q) parameter space is shown in Fig. 2.1. The complex eigenvalue A has a positive real part if either of the coefficients p or q is negative. We are interested in the nonoscillatory instability which leads to a new steady-state structure, realized when q < 0 or when \p\ < -2y/q for q > 0. The homogeneous state is stable for ft < 1. For /? > 1, on the other hand, it is unstable either to temporal disturbances where the eigenvalue exhibits a pair of complex conjugate roots, or to monotonically growing disturbances when one root appears on the positive real axis of the Aplane. The Turing spontaneous exchange of stability, which requires 0 > 1, occurs when Du 2d as in Fig. 2.6, on the other hand, viscous damping is primarily due to vertical gradients of those fields. The important control parameter for this flow is the Rayleigh number Ra = agATdP/nv, where a = (l/p)(dp/dT) is the thermal expansion coef-

Linear Stage of Pattern Formation

Z^

'/////////////
~ fc3/2.

An infinite number of wave pairs of frequency w p /2 whose wave vectors k(w p /2) have equal magnitude but opposite direction will be in resonance with the external excitation field. All wave vectors lie on the circle k^ + ky = k% where fco = {p^pl^l)2^In t n e absence of viscosity, all these pairs of waves will grow in time according to eat where the growth rate a is proportional to the amplitude Ap of the oscillating gravity field. If viscosity is taken into account as in the previous examples, the parameter Ap will have a threshold value above which one of many possible capillary patterns will be established. Again, important questions arise: What will occur after a sufficiently long period of time? Why do some mode pairs, which have merged together, survive and others disappear? Which governing parameters control the formation, for example, of a capillary crystal? These questions will be addressed in the chapters that follow.

Chapter 3

Model Equations

Observation always involves theory. Edwin Hubble What we see in the theory of pattern formation that is only now taking shape, is a situation reminiscent of what occurred in the theory of critical phenomena and in other developed branches of nonlinear physics. Many phenomena are being investigated by means of equations and models that, at first sight, do not appear to be relevant to the experiment described. For example, the basic notion in the theory of critical phenomena is the model Hamiltonian. Formally, it may be irrelevant to any particular physical situation, but the model is universal in that it describes effects common to a broad class of critical phenomena. Universality and simplicity make model equations a good tool in the construction of a qualitative theory. What do the words 'qualitative theory' connote? When faced with com­ plex phenomena that can be modeled as time-dependent nonlinear fields, the discovery of exact solutions usually represents a fortuitous event rather than genuine success, and it is certain that we shall not often have such luck. Qualitative theories are formulated to provide a more or less selfconsistent description of a class of physical events. The term qualitative is not meant to suggest the absence of rigor; it has the same meaning as, for example, in the qualitative theory of differential equations. A qualitative theory implies first the selection or construction of some basic nonlinear model relevant to a class of phenomena of physical origin. Usually the theory is devised in connection with the design of key experi­ ments for testing the universality of the model, for as far as we know there is no other way to validate the model. 27

28

Model Equations

The second component needed in the construction of a qualitative theory is the development of approximate methods, primarily asymptotic ones using various small parameters. The reader should be aware that the small parameter may not appear explicitly in the basic equations as a result of nondimensionalization, but instead may be determined by the properties of the solution. For instance, it may be connected with a relatively fast field decay at the periphery of some structures so that the interaction of different structures, determined by the value of their potential in the vicinity of other structures, may be considered as the small parameter. The third component for the development of a qualitative theory is computer simulation, without which it is often impossible to make sense of complicated problems of spatio-temporal pattern dynamics. The goal here is not merely to record some particular solutions, but rather to investigate the phenomenon as a whole. We emphasize the word 'phenomenon' and not system or model. We are interested not in the model per se, but in the effects it produces. For example, one may be interested in the chaotic scat­ tering of particles in soliton fields, or in the formation of finite-dimensional disorder as a result of temporal evolution. What kind of model should one choose? Must it correspond exactly to a particular physical system or may it differ in certain respects from this system? Sometimes the differences are of minor importance for the inves­ tigation of characteristic features of the phenomenon. In practice, a broad variety of models describing a given phenomenon does not reveal many qualitatively distinct features if the models are not sufficiently robust. We often 'violate' rational theories in order to draw on some truncated models. If we are able to advance our understanding of the processes within these models, we later might be able to complete them, if necessary, and obtain a more exact quantitative description. This illustrates the distinction be­ tween physical and formally mathematical approaches. A qualitative phys­ ical approach often yields definite conclusions for problems that at present cannot be solved rigorously. Such complex problems are often encountered by physicists, chemists, and biologists. The tendency to construct a qualitative theory is typical for funda­ mental branches of physics. In particular, R. Feynman (Feynman et a/., 1964) stated, perhaps a little pompously, that "The forthcoming great era of awakening of human intellect will bring about a method for the un­ derstanding of qualitative contents of equations. Today we are not able to do that. Today we cannot see in the equations for hydrodynamic flow

Swift-Hohenberg

equation

29

things such as the spiral structure of turbulence... We cannot say whether anything beyond the equations is needed." 3.1

Swift-Hohenberg equation

We will commence our discussion of model equations in the dynamical theory of pattern formation with one of its most popular paradigms, the Swift-Hohenberg (SH) equation du _ 1_ dt T0

§2^2

+

k

o)2+92U-93u2-g5u4 u.

(3.1)

Here, To and fo represent, respectively, the temporal and spatial scales of the process controlling the order parameter u(x,y,t). Equation (3.1) is often employed in the description of patterns in Rayleigh-Benard convection. In this context e > 0 is the distance above the critical value Rac for onset of instability, i.e. £ = (Ra/Rac) — 1 where Ra is the Rayleigh number, ko is the wavenumber of the mode that first becomes unstable, and the constant coefficients 52,3,5 measure the importance of second-, third-, and fifth-order nonlinearities. The characteristic spatial scales are measured in units of the fluid depth d and temporal scales in units of the thermal diffusion time T ~ d2 /K, where K is the fluid thermal diffusivity. It is remarkable that when the numerical values of all coefficients in Eq. (3.1) are chosen near the threshold, the SH equation gives a qualitatively correct description of the formation and evolution of convective roll patterns observed in laboratory experiments. In the following paragraph we attach a physical meaning to the nonlinear terms in the SH equation. When e > 0, the bifurcation is said to be supercritical since the onset of instability occurs for Ra > Rac. If in addition one has a so-called Boussinesq liquid for which 52 = 0, and it is appropriate to take into account only stabilizing cubic nonlinearity g3 > 0, then Eq. (3.1) may be represented in the traditional dimensionless form (Swift and Hohenberg, 1977) ^=[£-(V2

+ l)2-u2]u.

(3.2)

This form of the SH equation can be obtained, for example, from the Boussi­ nesq equations for convection in the limit of large Prandtl number. When £ < 0 and (73 < 0, the bifurcation is said to be subcritical since the onset of instability occurs for Ra < Rac. Such an instability can be realized

Model Equations

30

only if the perturbations are finite. In this case, fifth-order nonlinearity must be taken into account in the description of stabilization processes. Quadratic nonlinearity (52 ^ 0) comes into play when the fluid viscosity depends on temperature, or when the density exhibits a nonlinear depen­ dence on temperature (Haken, 1983); these are examples of non-Boussinesq convection. The model given by Eq. (3.1) describes pattern formation phenomena in initially homogeneous and isotropic media. This equation is invariant relative to translation and rotation. In the case g2 = 0 all solutions of the SH equation are symmetric with respect to reflection, u —► — u. In a subcritical bifurcation (e < 0), there exists a stable homogeneous state «(r, r) = 0. This trivial solution becomes unstable at e = 0 for arbitrary sinusoidal perturbations with wavenumbers |k| = fco, where fco = 1 in the nondimensional form of Eq. (3.2). In a supercritical bifurcation (e > 0), there exist stable, regular or nearly regular patterns for small e represented by periodic functions having wave vectors localized in a small neighborhood of the circle of radius fco, i.e. inside the annulus |fc —fco|< y/t- Owing to certain properties of the algebraic nonlinearity, the SH equation cannot possibly describe the entire gamut of patterns that may arise from arbitrary sinusoidal perturbations. This important issue will be discussed in more detail in Chapter 6 on quasicrystals. The SH equation can also be written in the general gradient form £

= /(«)-(l

+

V')V

/ H = - g

(3.3)

representing an equation with free energy functional F = jf [ [ / ( « ) + | ( ( 1 +

V »

dxdy

(3.4)

where U(u) is some potential and fi is the two-dimensional region of space in which pattern formation occurs. The existence of the functional F enables us to write Eq. (3.3) in a gradient form du

m

6F =

- ^

(3 0)

-

with variational derivative on the right-hand side. Note that the time vari-

Swift-Hohenberg

equation

31

F

P Fig. 3.1 Schematic showing minima of the free energy functional; P indicates different states in the configuration space of the system.

ation of F is given by dF

f \dU

„

„ , . 1 du , ,

Since the expression in square brackets is identical to the negative righthand side of (3.3) we have

f-/ a (!y**so.



The free energy in this problem is the so-called Lyapunov functional F which may only decrease as it evolves along its trajectory in some phase space. If the free energy functional has no minima, then when the horizontal scale of the liquid container is large compared to the instability wavelength, a propagating front will be observed as, for example, in chemically reacting flames. In this case, the free energy functional will decrease continuously until the front approaches the boundary of the medium. An alternative possibility is realized when the free energy functional has one or several minima as illustrated in Fig. 3.1, each corresponding to an equilibrium state in time. In this case multi-stability is possible. We conclude, therefore, that the limit behavior for gradient systems is characterized by either a steady attractor or propagating fronts. We will consider the case when the free energy functional has extrema, since steady patterns exist at each minimum. Spatial field distributions corresponding to different minima may be qualitatively distinct. For ex-

32

Model Equations

Fig. 3.2 Computer simulation of Eq. (3.1) showing the evolution of a hexagonal struc­ ture. The initial disturbances are (a) rolls, (b) a disk, (c) a square, and (d) seven symmetrically distributed circles. Taken from (Gaponov-Grekhov, et al., 1989). ample one may be periodic, another quasiperiodic, a n d others may exhibit

states of stable dynamical disorder. One example that illustrates the potentialities of the SH equation for describing real pattern formation is in Rayleigh-Benard convection. A

Swift-Hohenberg

equation

33

Fig. 3.3 Experimentally observed nucleation of hexagonal cells in slightly supercritical Rayleigh-B6nard convection; the lattice is spreading over the differentially-heated liquid layer in a large aspect ratio cavity r = 86 at Pr — 1. Taken from (Bodenschatz et al., 1991).

computer investigation of the SH equation with different initial conditions yields, in a two-dimensional geometry for e > 0 and 52 > {92)0 >n Eq. (3.1), hexagonal structures like the ones displayed in Fig. 3.2. The following interesting phenomenon was observed in those computer experiments. If a single hexagonal cell is taken as the initial condition, it rapidly 'accumulates' neighbors and builds up an entire 'crystal lattice.' This is similar to what is observed in laboratory experiments (Bodenschatz et al, 1991; Bodenschatz et al, 2000) on Rayleigh-Benard convection as shown in Fig. 3.3. Moreover, the same corroboration with laboratory ex­ periments is obtained from numerical simulations (Gaponov-Grekhov et al., 1989) for sufficiently large amplitude when the excitation is slightly below the threshold for convection, i.e. for a subcritical bifurcation where a fi­ nite initial perturbation is needed to exceed the excitation threshold. The results show that this perturbation grows, transforms into a universal struc­ ture, and exists as a localized state; see Chapter 8 on localized patterns. The evolution of the functional F in time may be readily tracked through computer simulation, and some of these results are presented in Figs. 3.4

34

Model

Equations

F(t)
J j , kx —> —igj, and ky —> —i-gz-, one obtains (in dimensioniess form) the linear part of Eq. (3.11). The NWS equation can be derived from the SH equation in certain restrictive cases; however, it can also be derived from more general primitive equations when compared to the SH equation. Here we establish directly that the NWS equation is a gradient model. Eq. (3.11) may be written in the gradient form 8A dt

SF 8 A'

(3.13)

with the free energy functional F =

-e\A\* +-\A\*

+

(JL _ i£.)2 \dx

A

dxdy

(3.14)

dy )

from which it follows that

d£ dt -

/

\dt\

dxdy < 0.

(3.15)

These potential (gradient) models may be used only for the descrip­ tion of relatively simple, monotonic dynamics of nonequilibrium media. If a physical phenomenon may be cast in the form of Eqs. (3.5) or (3.13), one may be sure that neither periodic pulsations nor chaotic temporal dy­ namics can evolve within the framework of this model. If such phenomena

Model Equations

38

are expected, then the model must be generalized. In particular, all re­ strictions concerning its gradient form must be removed. For example, if convective rolls are traveling rather than stationary, the coefficients of the NWS equation become complex and Eq. (3.13) is no longer correct.

3.3

Coupled amplitude equations

We may attempt to obtain a NWS equation not from Eq. (3.2) but from the more general model (3.1) when 52 = /3 and 35 = 0. Although this approach is feasible, it is not particularly effective. The point is that the NWS equation describes patterns against a background field of rolls. The rolls may have arbitrary orientation, part of the phenomena known as planform degeneracy, but it is assumed that all rolls have the same orientation. If the coefficient 0 of quadratic nonlinearity is not too small, the solution is unstable relative to the excitation of rolls making an angle of 0 = 120° to each other, giving rise to the formation of a hexagonal crystal lattice. Thus, in the derivation of a field envelope equation, the approximate solution must take into account all three roll systems of different orientations. Then, instead of (3.10) we assume the solution form u(x,y,t) = 5 > , ( * i ^ , T ) e i k ' - r + c.c.

0 = 1,2,3)

(3.16)

1

where |kj| = fco and the wave vectors satisfy the resonance condition 53j = 1 kj = 0, and (Xt, Vj) is an orthogonal coordinate system in the Ith system of rolls; see Fig. 3.7. The amplitude At again changes more slowly across the rolls than along them such that Xi = ex/2xi, Yi = exl*yi and we introduce the slow time scale T/ = et, where e 1 for j ^ /.

convection in a rotating system. The investigation of rotating convection is necessary for the understanding of the dynamics of stellar systems, mo­ tion in planetary atmospheres, and circulation in ocean basins. As has been shown by (Kiippers and Lortz, 1969), the originally stable pattern of straight rolls in a large system will lose its stability to another roll pattern at the angle 0 a 60° relative to the original one, when the rotation rate is greater than some critical value. The new set will, in turn, lose its stability to another roll pattern again at 0 w 60°, and so on; this means that there is no stable steady-state pattern for this case. Using this information (Busse and Heikes, 1980) proposed a dynamical model identical to Eqs. (3.17), but without spatial derivatives and with /? = 0. For rotating convection they found p i 2 = P23 = p3i = p+, whilst fai = Pz2 = Pu = P-, a con­ sequence of the rotational sequence of subscripts; but p+ ^ p- due to the lack of reflectional symmetry. The Kiippers-Lortz instability corresponds to p+ > 1 and p- < 1. For spatial homogeneity (d/dX = d/dY = 0) Eqs. (3.17) represent an ODE system which is still dissipative. For >3 = 0, the three ODE's have no stable fixed point, and the dynamics of this dy­ namical system is controlled by the heteroclinic orbit (separatrix contour;

Coupled amplitude

equations

41

Fig. 3.9 Heteroclinic loop for the spatially homogeneous form of Eqs. (3.17) with 0 = 0 and pij < 1, pji > 1.

see Appendix A.1.3) connecting the three saddle fixed points: ( e ^ ^ O ) , (0,£ 1/,2 ,0), and (0,0,e1^2) for pa — 1. This situation is illustrated in Fig. 3.9. Here, (|^4j|, \A2\, \Az\) is the vector of amplitudes of three sets of rolls with relative orientation 2^/3 radians to each other. Starting with an arbitrary initial condition, the dynamics will drive the system towards the heteroclinic orbit and the system will spend longer and longer time in the vicinity of each fixed point. A nonphysical feature of such a solution is that the properties of the system depend on the time elapsed since the initial conditions were set. Different resolutions of this paradoxical result are possible: one may take into account the resonant interaction of rolls having different orientations (/? > 0); one may include a spatial dependence of roll distribution; or finite noise may be added to the system (Busse and Hikes, 1980). For example, when ft > 0C, a stable limit cycle exists in the phase space of the corresponding system of ODE's as shown in Fig. 3.10. This implies that rolls having different orientations periodically replace one another in time. A strange attractor (see Appendix A.3.1) may also exist in restricted regions of parameter space in the framework of this model.

Model

42

Equations

Fig. 3.10 Limit cycle arriving in the neighborhood of the heteroclinic orbit shown in Fig. 3.9 for /3 > /3C, again with pij < 1, pji > 1.

3.4

Phase equations

Our reader may recall that in the derivation of certain models, one often tries to incorporate information on the character of the phenomenon under investigation. A restriction to some special class of solutions will likely provide a simplification of a set of governing equations that yields more detailed information about about the phenomenon of interest. The development of phase equations offers an important example of this process of model building. Suppose we are investigating the phase modu­ lation of a pattern governed by the NWS equation whose amplitude varies slowly in space and time. We begin by noting that Eq. (3.11) encompasses the family of spatially-modulated stationary solutions

A0(x) = y/i^f

ei(lx

(q2<e)

(3.18)

where A0(x) is depicted in either the first or fourth panel of Fig. 3.11, and q is the envelope wavenumber of the modulated x-rolls. We posit an unsteady envelope wave solution in the form A{x,y,t)

Ve

q2 +

<m

0i[qx+<j>(x,y,t)}

(3.19)

Phase initial pattern

43

equations

pattern with modulation

final pattern

Fig. 3.11 Development of an Eckhaus instability which may occur in either the forward (a,b,c,d) or reverse (d,c,b,a) direction; hence the wavelength of the final structure will be, respectively, either larger or smaller than the wavelength of the initial structure. Taken from (Manneville, 1990).

where off is the first order correction to the amplitude of the envelope field. Substitution of (3.19) into the NWS model (3.11) yields the linear phase diffusion equation dt

= D

d2 0, however, stable zigzag patterns such as those viewed in Fig. 3.12 are formed. It is important to emphasize that the above phase equations can work as model equations independent of the SH equation or the NWS equation. The phase diffusion equation can be derived using some general features of patterns like translational invariance of the periodic solution on the back­ ground of which we wish to investigate topological defects of the field; see (Berry, 1981) and (Newell et al., 1996).

Chapter 4

The Ginzburg-Landau Equation

...it is sometimes asked whether we live in a 'GinzburgLandau world'... Hugues Chat6 What form will a universal model describing the evolution of an or­ der parameter take after the principle homogeneous state has lost stability through an oscillatory Andronov-Hopf bifurcation? The experience gained with simple ODE systems shows that it will be an equation for a complex wave amplitude in the form of a Stuart-Landau equation (Stewartson and Stuart, 1971; Landau and Lifshitz, 1987) rIA

+ i0)\A\2A

^=eA-(l

(4.1)

wherein e is the supercriticality measuring the level of a control parame­ ter above its threshold for linear instability and /? is responsible for nonisochronous oscillations, i.e. the dependence of the oscillation frequency on its intensity. Taking into account the slow variation of complex mode am­ plitude on space and time gives, for a homogeneous and isotropic medium, the equation ^r-=eA-(l+ 0)\A\2A + (1 + ia)V2A (4.2) at in which a is the linear dispersion parameter. This complex GinzburgLandau (CGL) equation is one of the most popular models in the theory of pattern formation. A derivation of the CGL equation for a multicomponent reaction-diffusion system of a general form can be found in (Kuramoto, 45

46

The Ginzburg-Landau Equation

1984a). It describes the dynamics of chains or lattices of coupled nonlinear oscillators, pattern formation in two-dimensional chemical reactors where an autocatalytic reaction takes place (Kuramoto, 1984a), the dynamics of excitations in nerve membranes, the field evolution in nonlinear optical resonators (Moloney and Newell, 1990; Arecchi et al., 1999), structures in fluid flows (Newell and Whitehead, 1969; Segel, 1969), and many other physical processes. Below we consider two examples that illustrate how a CGL equation may be obtained in particular situations. First, however, we discuss some general properties of solutions of this equation. The past decade of avalanche-like popularity of the CGL equation were preceded by active and rather detailed investigations of its distinct limiting forms: for a = /3 = 0 one obtains the dissipative Ginzburg-Landau (DGL) equation 2£=eA-\A\iA

+ V2A

(4.3)

and if dispersion (ex a) and conservative nonlinearity (ex 0) are dominant, then one obtains the nonlinear Schrodinger equation - i ^ = sA-\A\2A + V2A. (4.4) at Eq. (4.4) has also been named the Gross-Pitaevsky equation in the study of vorticies in nonlinear fields (Pismen, 1999). Equation (4.3) was investigated at the end of the 1950's in connection with the problem of superconductiv­ ity (Ginzburg and Landau, 1950; Kupriyanov and Likharev, 1972; Gorkov and Kopnin, 1975). Stationary (Jj = 0) solutions of Eqs. (4.3) and (4.4) coincide, of course, but they may have quite different types of instability.

4.1

The dissipative Ginzburg-Landau equation

The DGL equation is a gradient equation dA

SF

ft"

6*

(45)

with free energy functional

F = -j^e\A\2-\\A\4

+ (VAf dxdy.

(4.6)

The dissipative Ginzburg-Landau

equation

47

By virtue of the fact that ^ = - / n | | £ \2dxdy < 0, solutions of Eq. (4.3) as t -> oo are either stationary field distributions satisfying, for e = 1, the equation V2A + A - \A\2A = 0

(4.7)

or fronts whose propagation is accompanied by a decrease of the functional F. The functional must reach its minimum at stable stationary solutions of Eq. (4.7). The most significant stationary solutions of the DGL equation include the so-called vortex solutions (Greenberg, 1980; Hagan, 1982), i.e. point defects or solutions corresponding to phase singularities of the field of A around which the circulation of the phase gradient is a multiple of 2n, viz. A(r,t) = R(T,t)e^ir't}

V 0 d l = 27rn

(4.8)

(n = 0 , ± 1 , ± 2 , . . . ) .

(4.9)

/ The quantity n is referred to as the topological charge of the vortex sin­ gularity. It is known that only vortices with \n\ — 1 are stable in the framework of the DGL equation. The isophase lines of a stationary vortex are straight lines emerging from the singularity. For complex coefficients in the DGL equation, the stationary point defect transforms into a spiral wave. A phase singularity with nonzero phase circulation around the vortex core also corresponds to a spiral wave. The isophase lines are no longer straight in this case, but resemble the spiral of Archimedes. The entire structure rotates around its central axis with constant angular velocity; see Chap­ ter 9 on spiral patterns. Such vortices and spirals are usually referred to as topological defects and the domain of the solution close to the singularity is called the defect core. Substituting solution form (4.8) into Eq. (4.3) gives two real coupled equations for the amplitude and phase, namely ^

= V2R +

{l-\Vf-R2}R

^ = V24> + 2R~1VRat

V(f>.

(4.10)

The Ginzburg-Landau Equation

48

Stationary vortices in polar coordinates (r, 9) with topological charge n have amplitude distributions Ro(r) described, for <j> = n6, by the equation dr2 +

I£

+

(i-i«-£)* =a

Mi,

Bearing in mind that \n\ = 1 is necessary for spiral stability, a straight forward exercise using coordinate expansions shows that the asymptotic behavior of the solution for large and small radii are

*oU~{b{1-&)

r->oc}

(412

)

where a and b are constants. Outside the core region, the vortex field intensity distribution rapidly tends to a fixed value. The dynamics of such defects will be considered in Chapter 9. Here we only mention that in the conservative model described by Eq. (4.4) the equations of motion of a point defect contain the so-called Magnus force and the solution is found by a simple hydrodynamic analogy. For details concerning this interesting feature see (Pismen, 1999, Chapter 4).

4.2

Nerve membrane excitation and the CGL equation

In the derivation of a CGL equation from full governing equations describ­ ing certain physical phenomena, special approximations may be invoked in each particular case; but it is always assumed that the field A(r,t) varies slowly in time and space. This yields in the evolution equation for A the additional Laplacian term V2A. This linear term implies that only nearestneighbor couplings are required for a medium to be considered as a lattice of individual elements. For example, the dynamics of the element of a nerve membrane can be described by the FitzHugh-Nagumo equations dv V3 e— = v - w — 3 dt dw = pv — qw

(4.13)

where the variable v is called the 'propagator' and w is referred to as the 'controller' (FitzHugh, 1961; Nagumo et al, 1962). Elimination of w in

Nerve membrane excitation and the CGL

Fig. 4.1 /i = 0.

Potential energy U(v)

Fig. 4.2

0v2/2 + P'v4/2

equation

49

of the nonlinear oscillator (4.14) with

Phase portrait for Eq. (4.14) showing a globally stable limit cycle.

Eqs. (4.13) yields the van der Pol-Duffing equation (see Table A.l in Ap­ pendix A) £il

- p(l - rf)$L + 0v + pi? = 0

(4.14)

where fi = 1 - eq, p = l/(i, /? = p-q, and 0' = q/3. For fi Ee^, P -> Pe**, and £>->£>. The dependent fields may be written in terms of a single complex order parameter A(x, y, t) E =

eA{x,y,t)e-^Kt

P = e(l +

i6)Ae-iURt

D = D0-e2\A\2.

(4.17)

Here e = \/DQ — Dc measures the distance above the threshold Dc of coher­ ent laser oscillation, UIR = (u>c + r?w a )/(l + 77), and 77 = K/JJ_ is the quality of the optical cavity. Substituting (4.17) into Eqs. (4.16) and averaging over the fast time yields the CGL equation ^

= ( l + T, + i»o\^\

(1 - Ml 2 ) A + i a ( l - i200r,X)V2A

where \ = I 1 + V + *o[(l ~ »?)/(! + v)}2}'1 and 0O = (wc - ua)/j± frequency mismatch parameter. Introducing the slow spatial

(4.18) is the

(?)-«(:) and temporal T = e2tn [(1 + T?)2 + {OoKr, - l)/fo + l ) ] } 2 ] " ' coordinates gives, in the limit 77 —► 0, the CGL equation ^

= (1 + iflo)(l - \A\2)A + m V 2 A

(4.19)

For the description of spatio-temporal dynamics of optical fields in active media we can employ, instead of Eqs. (4.2) or (4.19), a vector form of the CGL equation r)A ^ - = A - (1 + i/?)(A • A*)A -(6 + i'7?)(A • A)A* + ( 1 + m)V 2 x A (4.20)

52

The Ginzburg-Landau Equation

in which /?, S, 77, and a are real coefficients (Gil, 1993; Pismen, 1994; San Miguel, 1995). Various generalizations of the CGL equation are possible. For example, it may take into account nonhomogeneity of a medium which will result in the dependence of the coefficients on coordinates, such as beam inhomogeneity across the optical cavity, or the anisotropy of some nonequilibrium medium essential, for example, for a description of windgenerated waves (Rabinovich and Fabrikant, 1979). However, it should be emphasized that in its canonical form, the CGL equation (4.2) describes such a broad variety of phenomena in the dynamical theory of patterns that there is no need, as a rule, to go beyond the scope of this model when getting acquainted with the foundations of the theory.

4.4

Simple patterns in the CGL equation

The simplest patterns described by the CGL equation are coherent struc­ tures that depend only on the traveling coordinate £ = q • r — Sit. Such coherent structures can be localized or extended. An example is the plane wave solution A{r,t) = A.e*to--*-a{9'W

(4.21)

with A2 = 1 - q2 and il{qs) = (a - P)q2 + (3. The wavelength A is longer than 1-K since A = 2n/qs. If As varies in space or time, the local wavelength also varies because q2. = 1 — A2S. For q = q g + k with k ^ 0, the dispersion relation fi = fi(q) be­ comes complex. This means that the small perturbation proportional to = eA(k)t+ikr g r o w s a t t h e r a t e a _ R e (A(k)). After substituting A(r,t) 4 4e i[q..r-n(,.)t] + aeA(k)t+»k r i n t o E q (42) w ith e set to unity, one can find, for the most 'dangerous' longitudinal perturbations with k||q s , the dispersion relation A2 + 2{A28 + 2iqM

+ fc2)A + tfc{(l + 0 2 )(4g 2 - k2)ik

-2A 2 [(1 + ap)ik - 2{/3 - a)q}} = 0

(4.22)

in which (k — |k|). In the long wavelength limit k -> 0, we have A = ±ivgk - D||fc2 + 0{k3)

(4.23)

Simple patterns in the CGL equation

53

where vg -2{0-a)qs and D\\ = l+a/3 - 2 ( 1 + /32)(q2J(l - g 2 ) ) . One can see that these steadily propagating long waves are stable for D\\ > 0, i.e. the criterion for stability is

_iL_>_L±^_

(424)

The homogeneous oscillatory solution with fi = /? and qa = 0 is stable when a/3 + 1 > 0.

(4.25)

This is the Newell criterion for Benjamin-Feir (BF) instability. When the group velocity vg — &L = 2(/? — a)qs ^ 0, the BF instability has a convective nature and does not necessarily lead to destabilization of the traveling wave solution. It is easy to generalize the solution form (4.21) by supposing that both the phase and amplitude are propagating waves with different velocities such that A(T, t) = A(r - vt)e < ( q - r - n i ) .

(4.26)

Patterns of the form (4.26) can be quasi-periodic or localized soliton-like structures. The multiplicity of CGL patterns is perhaps most readily understood in its one-dimensional variant (V 2 = gjy)- This CGL equation has a family of so-called hole solutions (Nozaki and Bekki, 1985) in the form A{£, t) = [A(0 + rfvo]ei[HU)+ipVot-aH]

(4.27)

where A(£) = Atanh(jfcf),

~rr = Ktanh(ifef). a? Here, f = x — vot is a coordinate moving at the group velocity VQ of ampli­ tude modulation, A, K, and vo are real constants, r\ is a complex constant, and p = j ( a — /?). For f —► ±oo, these solutions tend asymptotically to the plane wave solutions

Mt, t) = yfi^Tj e-»t+i««

(j = 1,2)

(4.28)

with asymptotic wavenumbers qi (for f — ► —oo) and 172 (f° r £ —> +00) found from the dispersion relations Q=u(qj)

+ v0qj,

uJ(qj)=0

+ (a-p)q2j.

(4.29)

The Ginzburg-Landau Equation

54

From Eqs. (4.29) one finds w(gi) - v0qi = u{q2) - v0q2. This is the condition, in a frame of reference moving at constant velocity vo, of conservation of the constant phase difference between the asymptotic (£ -> ±oo) limits of the Nozaki-Bekki solution. Thus, the hole solutions are characterized by a localized dip in |A(x, t)| that moves with wavenumbers ql ^ q2 for nonzero velocity. However, the moving holes do not survive a general perturbation of CGL equation; such holes are structurally unstable. Therefore, only the case vo = 0 is physically interesting. In general, steady-state CGL structures in one dimension satisfy three coupled ODE's for real variables, obtained from Eq. (4.2) upon setting ^ = 0, and can be classified as pulses, fronts, or domain boundaries (Cross and Hohenberg, 1993). Domain boundaries are holes (sources) or shocks (sinks) representing heteroclinic solutions connecting plane wave states. A countable set of double-loop heteroclinic (see Appendix A.1.3) trajectories exists in the phase space of the ODE's (Bazhenov et al, 1996). These double-loops correspond to complex shock-hole-shock structures. In two dimensions, the zero amplitude topological defects with unde­ fined phase corresponding to vortices in the real GL decomposition (4.10), become rotating spiral waves in the CGL equation. An isolated spiral has the form A{r, 6, t) = A{r)e-Wt+me-*W

(4.30)

where (r, 6) are polar coordinates and m = ± 1 is the topological charge or circulation. The amplitude A(r) and phase ^ ( r ) have the asymptotic behaviors A(r) ~ r,

-^— ~ r dr .

Mr) ^VU^ti),

(r -¥ 0)

WO;

-fr^Q*

(r->oo)

(4.31)

where q, is the asymptotic wavenumber, a unique function of a and 0. The frequency of the spired rotation coincides with the frequency Cl — /? + (a — li)2q2t of the plane waves asymptotically emitted by the spirals. Again, the necessary condition for stability of the spirals is evidently D\\{q,) > 0; see Eq. (4.23). Well-developed spirals were found by (Aranson et al, 1991)

Phase equations

revisited

55

and (Chate and Manneville, 1996); an example of a spiral with shock-line defects is given in Fig. 4.5. Spirals can, in a sense, be looked upon as two-dimensional analogs of the one-dimensional standing hole solutions. The stability of spirals and spiral interactions will be considered in Chapter 9.

4.5

Phase equations revisited

As already discussed in Section 3.4, the slow space and time phase per­ turbations, or phase modes, play an important role in pattern dynamics. The now familiar examples from Chapter 3 are the zig-zag and the Eckhaus instabilities which appear on a background of stationary, spatially periodic patterns. For the general case of temporally oscillating patterns, the phase dynamics can be extremely rich and even chaotic. To exhibit this fact, let us construct an equation for the dynamics of phase modes based on the un­ perturbed, spatially-homogeneous, oscillatory solution of the CGL equation (4.2); this solution is Ao = Vee*0et+,».

(4.32)

The slowly modulated amplitude field can be written in the form A{ex, et) = [y/i + p{ex, e t)] e #»+*(«.'OI

(4.33)

where 17 = e/3. We wish to find equations for

k

i+N = ~ki>

A

J+" = ~A*i

(5-J)

where kt are basis vectors of the crystalline lattice, all of which lie on the circumference |kj| = k0 = 1 and making angles with neighboring wave vectors, and 6(k) is a vector delta function. Only N = 1, 2 and 3 correspond to crystals for two-dimensional patterns. Any other value of N gives rise to quasicrystals (see Chapter 6). The case N = 1 corresponds to a roll pattern, N = 2 corresponds to lattices with square cells, and N = 3 corresponds to a hexagonal lattice. It is convenient to use an evolution equation in the spectral form of the SH equation (3.1)

^

= [e - (k2 - l)2]Ak -0 J' Akl Ak2S(k - fc - k2) dhdk2

- f p(e)AkiAk2Ak36(k

- ki - k2 - k3) dkldk2dkz

(5.2)

devoid of quintic nonlinearity (#5 = 0). Here k and r are two-dimensional vectors lying in the plane of modes forming a pattern, k = |k|, 8{x) is the scalar delta function, Ak = A*_k are complex amplitudes of the e±lkT modes, and e is the supercriticality parameter. Also, 0 is the parame­ ter of square nonlinearity that determines a three-mode interaction of the form ki + k2 = k, p(Q) is the coefficient for the four-wave interaction ki + k2 + k3 = k determined by cubic nonlinearity, and 0 is the aperture angle of the rhombus composed of vectors k, k i , k 2 , and k 3 . For a homo­ geneous, isotropic medium without rotation, as in the model described by Eqs. (3.17)), p(6) satisfies the symmetry conditions p(Q) = p(-Q)

= p(Q + 2nm)

(5.3)

where m is an integer. Let us first consider in more detail the case when Ak is a Fourier transform of the order parameter described by an SH equation. Then Eq. (3.1) immediately implies p(Q) = canst = p(0).

(5.4)

'Crystal'

Formation

65

6" Fig. 5.1

Phase portrait for a two-mode competition described by Eq. (5.5).

In the absence of square nonlinearity when 0 = 0, the complex amplitudes of an arbitrary pair of spatially homogeneous modes are described by the coupled pair of ODE's ^

= e A,, 2 - | [\AU2\2 + 2|A 2 ,,| 2 ] Alt2

(5.5)

derived from Eq. (5.2) assuming u(r,t) = [Ai(t)eik,'r + A2(t)eik*r] + c.c, with |lci ,21 = 1- The phase portrait of Eq. (5.5) shown in Fig. 5.1 is com­ prised of, besides the unstable state A^ = A2 = 0, a saddle at Aj = A2 = 2%/2£/3 and two stable nodes: Ax = 2^/2^/3, A2 = 0 and Ax = 0, A2 = 2 \j2e/Z. This portrait describes the well-known effect of mode com­ petition; here only a single-roll regime is stable and all other perturbations are suppressed. The phase portrait of the three-mode competition was given in Fig. 3.8. Unfortunately, it is difficult to schematically represent the phase space for the dynamics of iV-mode competitions for TV > 3 described by the system

dA

«£,

— L = eAj; - Aj 2_, Pji | Aj | 2 ,

pji = pij,

pji > pjj.

(5.6)

1*3

These equations are the simplest form of the coupled amplitude equations (3.17). For the SH equation (3.2), we have in Eq. (5.6) pji = 3/4, and also Pa = 3/8. Using the expression for free energy, however, it can be readily verified that the regime of co-existence of rolls having different orientations

'Crystal'

66

Formation

is unstable irrespective of the number N. Indeed, from the expression for free energy F = - * £ > / + i £ f > | ^ | W i=\

(5.7)

1=1 j=\

it follows that in the single-mode regime 1 e2 F = F1 = --—. 2

Pjj

(5.8)

Furthermore, in the regime of simultaneous generation of N modes having intensities \Aj\2 = (e/ 5 3 I = 1 Pji), one finds N e2 FN = ~T=A—.

(5.9)

However, in this case the inequality (l/N) 5Z/=i Pji > Pjj must be satisfied; consequently, F# > F\. Thus, a simple crystal structure in the form of a periodic system of rolls is always established as a result of evolution in the framework of the gradient model of Eq. (5.6). The arbitrary orientation of the rolls depends only on initial conditions. When /? ^ 0, a simple roll structure may be unstable with respect to the excitation of a pair of modes which meet the condition of three-mode resonance: ki +k2 + k 3 = 0. For example, when non-Boussinesq effects are essential for convection, an instability that produces the birth of a crystal having third order rotational symmetry has been observed in an experiment by (Koschmieder, 1992); see Fig. 5.2. It is remarkable that the birth of a hexagonal lattice is very robust when the supercriticality is large. Results of computer experiments demonstrat­ ing different scenarios of the birth of hexagonal crystals in the framework of model (3.1) are presented in Fig. 3.2. One can see various intermediate states: rolls, targets, localized hexagons, etc. However, all are unstable and, as t —► oo, a space-filling hexagonal lattice state corresponding to the minimum free energy of Eq. (3.1) is established. The orientation of the hexagonal lattice depends weakly on boundary conditions in a horizontally extended system and is different in different experiments. If the supercriticality is increased, then defects in the per­ fect crystal structure appear; these defects form sections of the hexagonal

'Crystal'

Fig. in a mm cm.

Formation

67

5.2 Shadowgraph picture of a hexagonal lattice in a Benard-Marangoni convection 1.9 mm layer of silicone oil heated uniformly from below, residing underneath a 0.4 layer of air which sits below a uniformly cold sapphire lid; the cell diameter is 10.5 Taken from (Koschmieder, 1992).

lattice with different orientations like that shown in Fig. 2.3. A more com­ prehensive discussion of the formation and evolution of these pieces, called domains, will be given in Chapter 7. Mathematically, the modal structure of a hexagonal lattice (N = 3) is described by an ODE system of the form (3.1) with g$ = 0, namely dAj = eAj - P'A;A* - Aj dt

(5.10)

where i j= j ^ I. For not too small /?', this system has a single stable equi­ librium state A\ = A2 = A3 = AQ that corresponds to a regular hexagonal lattice. Naturally Eq. (5.10), like the SH equation, is a gradient system of the form (3.13) with functional

~/{

e(\A!|2 + \A2\2 + \A3\2) - 2 ( | ^ | 2 | > 1 2 | 2 + \At\2\A3\2

4 4 ■i(|A,| + |A 2 | + |A 3 | ) - p'{A\A'2A*3 + c.c.)\dxdy.

+ |A2|V3|2)

(5.11)

68

'Crystal' Formation

Fig. 5.3 Dependence of the interaction coefficient on rhombus aperture angle 0 when there is a coexistence of different modes.

Within this model, the formation of a crystal lattice with third-order sym­ metry must take place monotonically in time without oscillations, a fact confirmed by both physical and computer experiments. The birth of a hexagonal lattice in the framework of models (3.1) or (5.10) is an example of the birth of a crystal in which not only the am­ plitudes but also the phases of the modes forming the crystal are coupled; these will be referred to as hard crystals. It should be apparent that for the coexistence of disoriented modes which may form a crystal, there is no stringent requirement that some modes should generate other particular ones. It is essential that the modes be mutually exclusive and not interfere with the existence of other modes. In other words, there must be no strong competition. Is this possible in principle? The formal answer is simple. If the matrix elements pji are such that Pjj > Pji

(1*3)

(5.12)

or, more generally, if the coefficient p(Q) in Eq. (5.2) depends on 0 , as for example in Fig. 5.3, then the regime of joint generation of disoriented modes will be stable. In this case we have FN < F\\ see Eqs. (5-8) and (5.9). The phase portrait of the amplitude equations describing the twomode interaction described by Eq. (5.12) is presented in Fig. 5.4. One can see that simple patterns in the form of rolls are unstable and the regime \Ai\2 = Ci and \A2\2 = C2 corresponding, for instance, to the rhombic crystal exhibited in Fig. 5.5, will be established in the course of evolution from arbitrary initial conditions. These crystal structures may be called soft crystals. Unlike hard crystals, the phases of the modes forming

'Crystal'

Formation

69

Fig. 5.4 Phase portrait of a two-mode interaction described by the spatially homoge­ neous form of model (3.17) for 0 = 0 with pij < 1.

Fig. 5.5 Schematic of a soft crystal. The angle between the two wavenumber vectors is arbitrary.

a soft crystal are uncoupled and therefore arbitrary. This formal reasoning for the absence of strong competition, however, says nothing about the physics of the phenomenon. For instance, why do elementary modes sometimes compete strongly and yet at other times behave independently?

70

'Crystal' Formation

The explanation may be most easily understood in terms of energy arguments. Directly from Eq. (5.6) it follows that the phenomenon of com­ petition is the result of the additional nonlinear damping that one mode produces for another. If there is only a small energy dependence between the modes, then the additional damping is insignificant. This occurs, for example, in Rayleigh-Benard convection at high Prandtl number when the fluid thermal conductivity is weak. In this situation the SH equation is not correct and the nonlinearity responsible for the stabilization of linear instabilities must be supplemented with a term depending not only on the amplitude of the perturbation, but also on the gradient of its inhomogeneous spatial distribution. These may include terms of the form u(Vu)2, u(V 2 u) 2 , u(V 4 u 2 ), etc. The choice of the term depends on the inherent physics of the problem. At high Pr, for example, nonlinear dissipation may be written in the form u 3 - 6uV 4 u 2 . Then, instead of Eq. (3.1) we have (Bestehorn, 1992) § i = [ £ - ( V 2 + Jfc 2 )]u- U 3 + feUW. (5.13) L at This model is also employed for the description of convection in binary mixtures. Direct substitution of u = (A\etkx + A2e%ky) + c.c. representing the amplitudes of mutually orthogonal roll structures into Eq. (5.13) yields, on averaging and for b > 0, an equation of the form (5.6) with coefficients Pji that satisfy condition (5.12). The dissipation described by the term with coefficient b contributes only to self-action and thus diminishes the competition between roll structures. The time evolution of a random initial distribution of the field u(x, y, t) according to Eq. (5.13) is shown in Fig. 5.6 where it is seen that a crystal lattice with square cells is established even in a circular cavity at sufficiently large time (Bestehorn, 1992). Such a crystal with square cells was observed near the instability threshold for convection in a horizontal liquid layer at high Prandtl number (Le Gal et al., 1985). The resulting instability pattern is reproduced here in Fig. 5.7. We have considered two possible dynamical mechanisms of crystal pat­ tern formation. One is connected with a weak competition that allows for the coexistence and stability of the simultaneous generation of mode struc­ tures. Such crystals as labeled soft. The other mechanism is based on the resonant interaction of modes when they not only co-exist without interfer­ ence but also stimulate each another. These crystals are labeled hard and examples include square and hexagonal cell lattices.

'Crystal'

Formation

71

Fig. 5.6 Appearance of a square pattern in a computer simulation of Eq. (5.13). Taken from (Bestehorn, 1992).

Fig. 5.7 A stationary square pattern in thermoconvection at Pr = 70 just above the instability threshold at e = 0.02. Taken from (Le Gal et al., 1985).

The crystalline patterns discussed above have a very simple spatial Fourier spectrum. What is the spatial planform of a two-dimensional pat­ tern when the power spectrum is more complex but discrete? To answer this question we use analogous results obtained from time series data. It is known that a discrete Fourier spectrum does not ensure that the cor­ responding time series is periodic, or even predictable at large times. If the spectrum consists of several incommensurate frequencies, the Poincare return time for the corresponding quasi-periodic signal may be so long

72

'Crystal' Formation

Fig. 5.8 A superlattice pattern in capillary ripples simultaneously excited by two fre­ quencies. Taken from (Kudrolly et al., 1998).

♦ K.

Fig. 5.9 Two-dimensional power spectrum of a superlattice. The two sets of circles correspond to the two hexagonal lattices of the same wavenumber. Taken from (Kudrolly et al., 1998).

that any finite time series of the signal would appear random (see Ap­ pendix A.1.5). Further discussion on this point is given in Section 6.3. When the power spectrum has no incommensurate frequency, the time se­ ries is periodic; but it may still be quite complex as in the beating between several different harmonic waves. Returning to our consideration of spatial planforms, one can expect in nonequilibrium media to see quasi-periodic spatial patterns and complex spatial patterns made up of simple lattices. The quasi-periodic patterns are quasicrystals to be discussed in the next chapter. Here we focus on crystalline modulations of crystal patterns: supercrystals or superlattices.

'Crystal'

Formation

73

The discrete spatial power spectra of these space-filling crystals can be quite complicated. An example of a superlattice pattern generated by twofrequency forcing of a free surface liquid layer is given in Fig. 5.8. Gollub and collaborators (Kudrolly et ai, 1998) named this pattern a superlattice because it is composed of two hexagonal sublattices. Figure 5.9 shows that the spatial power spectrum of this superlattice contains peaks at wavenumbers smaller than those observed at the onset of instability. The dominant twelve peaks in this figure are not equally spaced on the circle of radius ko and may be described as two sets of six peaks, each set forming a hexagon oriented at an angle 6 « 22° to one another. This critical orientation may be explained as follows. Suppose the wave vectors of plane waves forming one hexagonal sublattice are ki = (1,0), k 2 = ( - | , ^ ) , and k 3 = ( - 5 , - ^ ) oriented at an angle 6 = 120° to each other. If another hexagon with wavevectors k'1; k'2, and k'3, rotated through the angle 9 ss 22° is also present, then the combined pattern has to be periodic because of the commensurability condition 2k'j — k 3 = ki — 2k,3. These supercrystals have a long lifetime which can only be explained by investigating the stability of the phase-locking regime for the coexistence of the two modes comprising the superlattice. This problem is mathematically too complex to discuss here; the interested reader may consult (Silber and Proctor, 1998) for details.

Chapter 6

Quasicrystals

This number is nothing but the ratio known as a "golden section" that plays an extremely important role in an at­ tempt to reduce the beauty of proportions to a mathemati­ cal formula. Hermann Weyl Let us project a periodic structure in the form of a lattice of M-dimensional cubic cells onto a plane of lower dimension m. For M = 5 and m = 2 we will obtain for most angles of projection the irregular pattern, fascinating in its beauty, shown in Fig. 6.1. This is the famous Penrose tiling built of rhombi of two types having acute angles of 36° and 72° (Penrose, 1974, 1979). The Penrose tiling is sometimes called a geometrical toy but such a crystal with tenth-order symmetry is observed in Nature. We are now aware of these two-dimensional quasicrystal structures with fourth-order symmetry in thin films of alloys, with fifth-order symmetry in laminated substances, and with sixth-order symmetry in small solid particles. The well known 'algebraic' model of a two-dimensional quasicrystal represents the spectral distribution of material density in the form 2N

under the condition that the projections of the vectors kj onto any axis are mutually incommensurate. A base of five standing waves with the angles of 72° between neighboring wave vectors gives birth a to Penrose tiling. Like ordinary crystals, the pattern in Fig. 6.1 possesses long-range order, which 75

76

Quasicrystals C"*'i %fc

£ / 7>)

^L—*"f *~-^^

\i JL\

^■•jf \Py

V^T^-j

O*-*/^'*"^^

)C

Fig. 6.1 Two views of a Penrose tiling; the pattern in (b) has twice the magnification of the pattern in (a). Taken from (Nelson, 1986).

follows at least from the fact that this picture is obtained as a projection of a perfect crystal. Penrose tilings, however, also possess local symmetry relative to rotation about an axis by an angle of 72° which is no longer compatible with the translational invariance needed for crystals. Generally, the local symmetry is determined by invariance through a rotation of angle TT/N for integer N, and here TV = M = 5. We draw the reader's attention to the fact that arbitrarily large fragments with fifth-order symmetry may be found in the photograph in Fig. 6.1. If the scales of successively larger structures in Fig. 6.1b are carefully compared, the origin of this beauty of proportions can be readily understood: the scales differ by a factor /x = {y/% — l ) / 2 and hence /x is the golden mean or golden section. The spatial Fourier spectrum of a quasicrystal observed in real life differs only slightly from that in Eq. 6.1. It contains not only N basic modes whose projections are incommensurate, but also their higher and mixed space harmonics IN

u(r) = 2_j Ajelilj

r

+ (higher and mixed spatial harmonics).

(6.2)

i=i

The quasicrystal spectrum may contain an infinite number of discrete com­ ponents, possibly arranged arbitrarily dense forming a Cantor set. But there are sometimes only a few specific peaks corresponding to periodic modes forming quasicrystals. Such peaks have been observed in experi­ ments on Bragg scattering in metallic alloys (Shechtman et a/., 1984). Although over the past two decades there have appeared a great num­ ber of experiments where quasicrystalline patterns were observed, all were concerned with microphysical objects, primarily metallic and intermetallic

Quasicrystals

77

Fig. 6.2 Computer generated image of an eightfold quasicrystal pattern. Taken from (Christiansen et al., 1992).

alloys. Quasicrystal order in macrophysics was observed for the first time by Danish researchers (Christiansen et al., 1992). These were stable qua­ sicrystal patterns composed of capillary ripples on the surface of a fluid layer in an oscillating gravitational field (see Appendix B.l for a discussion of this laboratory experiment). The Danish team found stable octagons, i.e. eighth-order symmetry quasicrystals formed by TV = 4 standing capillary waves with 45° intersecting angles as shown in Fig. 6.2. These numerical simulations are to be compared with the viscous fluid experiment shown in Fig. 1.6 performed in a cylindrical cell of large diameter d ~ 50A, where A = 2ir/ko is the wavelength of parametrically excited waves. In France (Edwards and Fauve, 1993) observed a dodecahedral qua­ sicrystal formed by six standing waves making an angle of 30° with one another. For the different containers shown in Fig. 1.7, one can see that the form of quasicrystalline patterns was the same in a circular cell and in a cell with sidewalls in the shape of France for which the distance between Bordeaux and Geneva was 5 cm. The characteristic feature of that experi­ ment was biharmonic excitation resulting in the simultaneous formation of modes with wavenumbers ko and fcn + 6k, where Sk/ko ss 0.17. The conse­ quences of this interaction are of major importance and will be considered later.

78

Quasicrystals

Fig. 6.3

6.1

Wavenumber vectors of interacting modes in the (kx,ky)

plane.

Octagons, decagons, and dodecagons

Let us consider mode superpositions (6.1) from a dynamical point of view in order to find the evolution of crystal and quasicrystal patterns with a discrete set of 2N Fourier components. Using Eq. (6.2), and taking into account only quadratic and cubic nonlinearity of the media, one can derive from the Swift-Hohenberg model (3.1) a set of equations for the amplitudes of these modes in the form dAj dt

N 7o-4p(O)|Aj|

2

-8^p(0j/)|ylj|2

Aj + 0 7

,Ai+2iAj+il.

J=I

(6.3) Here Ojt = n(j — l)/N is the angle between the vectors kj and kj, and p(0) = p(—Q) = p(Q + 27rm) as before in Eq. (5.3). The interaction term proportional to 0' is present for certain angles between the wave vectors; /?' = 0 only when there exists an integer / = iV/3. Moreover, here we consider only modes for which the angle between the wave vectors is much larger than 0(e1/2) which is the relative width of the packets of interacting modes, i.e. the width of the ring in Fig. 6.3. For N = 4 (0' = 0) the dy­ namics of system (6.3) is characterized by the two independent parameters 7i = 2 p(*/4) P(0) '

72=2

Pfr/2) P(0) '

Octagons, decagons, and dodecagons

79

These TV = 4 quasicrystals are octagons corresponding to the equilibrium state of Eqs. (6.3) given by

1*1 = 1*1 = 1,131 = |*| = ]JYT^TT2-

(6-4)

Such patterns may be seen in Fig. 6.2. Analysis of the stability of this state gives the needed criteria for the stability of octagons, namely 7i < (1 + 72)/2,

72 < 1.

(6.5)

A complete analysis of octagonal pattern stability relative to arbitrary per­ turbations must be performed taking as a basis Eqs. (6.3) linearized in the neighborhood of the solution (6.4). This analysis is very cumbersome, so we refer the interested reader to (Malomed et a/., 1989) for details. The quasicrystal with N = 5 (/?' = 0) depicted in Fig. 6.1 is composed of decagons. It can be described, like octagons, only by means of Eqs. (6.3) with P' = 0. In this case we have the independent parameters o pfr/5) p(0)

0

P(2TT/5)

p(0)

This quasicrystal structure corresponds to the equal mode amplitude equi­ librium state

1*1 = 1*1 = 1*1 = 1*1 = 1*1 = y/1 + 2£+2ja

(6-6)

with arbitrary phases. One can also easily determine within Eqs. (6.3) the following necessary conditions for stability 1 + V7i - M72 > 0,

1 + ^72 - M7i > 0

(6.7)

where p = 2cos(7r/5) is the golden mean and ip = 2cos(27r/5). The computer generated quasicrystal pattern with N = 6 (/?' = 0) displayed in Fig. 6.4 are composed of dodecagons. Like quasicrystals with lower symmetry (N — 4,5), these quasicrystals are composed of modes of arbitrary phase. For /?' ^ 0, however, experimentally observed quasicrystal structures (Edwards and Fauve, 1993) no longer represent a composition of standing waves, but of hexagons which may be considered as 'normal' patterns in this case; the explanation is as follows. Inside of each hexagon there is a

80

Quasicrystals

Fig. 6.4 Computer generated image of a twelve-fold quasicrystal pattern. Taken from (Edwards and Fauve, 1993).

three-mode resonance ki + k2 + k3 = 0, the phases <j)j of which are related by

sin I Yi i J = 0

(6-8)

where j is even for one sublattice wave triplet and odd for the other. Of course, each hexagonal lattice can be considered as a composition of rolls. However, in this case (/?' ^ 0) the rolls form a hard crystal, and when we consider the interaction between two different hard crystals corresponding to different forcing frequencies, these crystals play the role of nonlinear normal modes. We have in Eqs. (5.2) or (6.3) square nonlinearity which is associated with the mechanism of explosive (faster than exponential) instability. It is significant that an explosive instability leads to the excitation of structures, and in fact this is a common mechanism for the excitation of hexagons. Let us return again to the experiment on capillary quasicrystals with N = 6. As pointed out at the end of Chapter 5, dodecagons may be consid­ ered as a composition of two hexagons which have almost equal amplitudes

A generalized Swift-Hohenberg

model

81

and make an angle of 30° with one another. The necessary condition for the stable coexistence of such hexagons is weak mode competition. A pos­ sible way to realize this in the laboratory is to supply energy to different hexagons from different sources. This is exactly what was done in the capillary experiment of (Edwards and Fauve, 1993) which generated the quasicrystal pattern in Fig. 1.7; here, the vertically vibrating liquid layer was forced by two distinct, but nearly equal frequencies.

6.2

A generalized Swift-Hohenberg model

Quasicrystals in two-dimensional nonequilibrium media can be constructed not on the basis of modal equations like Eqs. (6.3), but by incorporating a rotationally invariant model equation. Such a generalization of the SwiftHohenberg equation suggested by (Lifshitz and Petrich, 1997) has the form 5 ^ = eu + au2 - u3 - c(V 2 + 1)(V 2 + q2)u. at

(6.9)

Here the two operators (V 2 -I- 1) and (V 2 -I- q2) describe the excitation of patterns with two different space scales, each corresponding to a different forcing frequency. The wavelength-selecting term in Eq. (3.1) is replaced by a similar term here which damps out all modes except those near wavenumbers fco = 1 and k'0 — q. Thus this model admits two unstable length scales and retains the square nonlinearity responsible for triad interactions among the different modes. Equation (6.9) is a gradient model and stable quasicrystalline patterns correspond to minima of the Lyapunov functional. If, in the control parameter space where a quasicrystal exists, other stable patterns are absent, this minimum must be global. Lifshitz and Petrich calculated the Lyapunov functional of rolls, hexagons, and dodecahedral patterns for q = 2cos(7r/12). They found the values of the functional for rolls

- —-e V pF 2 — o for hexagons

2

82

Quasicrystals

and for dodecagons A0\3 /

r

67F\

20 A

2 r

67F\

,

9 ,2

whose Fourier spectrum contains two twelve-fold stars of wave vectors, one on the circle fco = 1 and the other on k'0 = q. Here e*2 = e/a2. From the above results one finds that for e* > 1.913 rolls have the lowest free energy; for 1.913 > e* > 0.088 hexagons are observed; and for e* < 0.088 dodecahedral quasicrystals will be selected. Of course this model is oversimplified for the description of pattern evo­ lution in a particular experiment such as the Faraday problem. However, the model does provide a clear understanding of the role of square nonlinearity on two-scale excitation for the formation of quasicrystals. 6.3

The 'turbulent' crystal

Quasicrystals with N > 4 without phase locking, which we call soft qua­ sicrystals, are the simplest quasiperiodic spatial patterns which contain arbitrary parameters — the phases of the modes. Let us recall here the well known Landau-Hopf model of turbulence (Landau, 1944; Hopf, 1948). Within this model, turbulence is represented as a set of modes with in­ commensurate frequencies and arbitrary phases. The corresponding time series N

u(t) = ^2 Aje^+O^

+ c.c.

(6.10)

i=i

looks absolutely random at sufficiently large N and is quite suitable for modeling irregular and unpredictable turbulent motion. The correlation function of the temporal series (6.10) decays rapidly and becomes equal to unity again only in a return time T that is very large (see Appendix A.1.5): this Poincare return time T ~ eaN (a ~ 1) grows very fast with increasing N (see Landau and Lifshitz, 1987, Chapter 3). Such a model of random temporal fluctuations, the mathematical image of which is an open winding on an TV-dimensional torus in the phase space of a dynamical system, could actually be valid; however, mathematical investigations have shown that quasiperiodic motions of a dissipative dynamical system with N > 3 are unstable as a rule (Newhouse et al., 1978). Either the open winding on

83

The 'turbulent' crystal

the torus closes forming a periodic attractor (limit cycle), or the TV-torus is destroyed and a new (strange) attractor is born. This attractor is a mathematical image of stable chaotic oscillations generated by a dynamical system; see Appendix A.3 for details. We have discussed the coexistence of incommensurate temporal modes. What happens in the spatial case? The stable coexistence of six modes (twelve waves) looks quite possible. What happens if N is increased sub­ stantially? Long-range order will be observed only at very long distances 0(eaN), so that the concept of a Poincare return length analogous to the Poincare return time (cf. Appendix A.1.5) becomes physically meaningless. Thus quasicrystalline structures will degenerate into a chaotic or 'turbulent' distribution of order parameter in space which may be called a turbulent crystal (Newell and Pomeau, 1993). Consider the evolution of order parameter u(r, t) determined by the variational model with free energy functional in the form (Newell and Pomeau, 1993) F = / { ^ [ ( V 2 + 1)«]2 - \eu2 + a ! ( V 2 « ) 4 + a 2 ( V U ) 4 F = J{\[&2 + l)u]2 - l-eu2 + a ! ( V 2 « ) 4 + a 2 ( V u ) 4

+ ^ u l l

+ -fu

2

+ fMTrnUrnU2 + \ £ 2

m

+ nrmumu

[(^

+ \ £[(V

2

+