nonlinear science at the dawn of the 21st century

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Nonlinear Science
at the Dawn
ofthe21stCentury
13
Editors
P. L. Christiansen
M. P. Sørensen
A. C. Scott
Department of Mathematical Modelling
The Technical University of Denmark
Building 321
2800 Kgs. Lyngby, Denmark
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Rememb ering Bob
Alwyn Scott
In the summer of 1962, young Robert Dana Parmentier was finishing a
master’s thesis in the Department of Electrical Engineering at the Univer-
sity of Wisconsin, where it had been decided to support a major expansion
of laboratory facilities in the rapidly developing area of solid state electron-
ics. Jim Nordman and I—both spanking new PhDs—were put in charge of
this effort, and we soon found ourselves involved in a variety of unfamiliar

activities, including the slicing, polishing, cleaning, and doping of semicon-
ductor crystals prior to the formation of p-n junctions by liquid and vapor
phase epitaxy in addition to the more conventional process of dot alloying.
We had much to learn, and welcomed Bob as a collaborator as he worked
toward his doctorate in the area.
It was an exciting time, with research opportunities beckoning to us from
several directions. From a more general perspective than had been origi-
nally contemplated by the Department, we began studying—both exper-
imentally and theoretically—nonlinear electromagnetic wave propagation
on semiconductor junctions with transverse dimensions large compared to a
wave length. And there were many interesting nonlinear effects to consider.
Using ordinary reverse biased semiconductor diodes, the nonlinear ca-
pacitance of the junction causes shock waves, suggesting a means for gen-
eration of short pulses. At high doping levels, the junctions emit light to
become semiconductor lasers, and at yet higher doping levels the negative
conductance discovered by Leo Esaki appears, leading to a family of trav-
eling wave amplifiers and oscillators. In 1966, this latter effect was also
realized on insulating junctions between superconducting metals, rendered
nonlinear through Ivar Giaever’s tunneling of normal electrons.
As a basis for our theoretical work, we started with John Scott Russell’s
classic Report on Waves, a massive work that had been resting on a shelf
of the University Library for well over a century, and in 1963 two events
occurred that were to have decisive influences on Bob’s professional life.
The first of these was a Nobel Prize award to the British electrophysiolo-
gists Alan Hodgkin and Andrew Huxley for their masterful experimental,
theoretical and numerical investigations of nonlinear wave propagation on
a nerve fiber. This seminal work—to which applied mathematicians made
no contributions whatsoever—pointed the way to Bob’s doctoral research
on the neuristor, a term recently coined for an electronic analog of a nerve
axon.

The other event of 1963 was the experimental verification of Brian Joseph-
son’s prediction of tunneling by coupled electron pairs between supercon-
ducting metals, leading to an unusual sort of nonlinear inductor for which
iv
current is a periodic function of the magnetic flux. From this effect, the
relevant nonlinear wave equation for transverse electromagnetic waves on
a strip-line structure takes the form
∂
2
φ
∂x
2
−
∂
2
φ
∂t
2
= sin φ, (0.1)
where φ is a normalized measure of the magnetic flux trapped between the
two superconducting strips.
Originally proposed in 1938 to describe dislocation dynamics in crystals
and later to become widely known as the sine-Gordon equation, this is a
nonlinear wave equation that conserves energy (which nerves and neuristors
do not), and by the spring of 1966 we were aware that it carries little lumps
of magnetic flux very much as Scott Russell’s Great Wave of Translation
transported lumps of water on the Union Canal near Edinburgh.
Just as Equation (0.1) can be viewed as a nonlinear augmentation of the
standard wave equation, the system
∂

2
u
∂x
2
−
∂u
∂t
= u(u − a)(u − 1) , (0.2)
is a nonlinear augmentation of the linear diffusion equation. Originally
proposed in 1937 to describe the diffusion of genetic variations in spatially
distributed populations, Equation (0.2) is the basic equation of excitable
media, now known to have a variety of applications in chemistry and bi-
ology. Since it has a nonlinear traveling wave solution that represents the
leading edge of a Hodgkin–Huxley nerve impulse, this equation is of central
interest in the theory of a neuristor.
From a broader perspective, Equation (0.1) describes basic features of
nonlinear wave propagation on closed (or energy conserving) systems, while
Equation (0.2) plays the same role for open (or energy dissipating) systems;
thus the two equations are fundamentally different and their traveling wave
solutions have quite different behaviors. Equation (0.1) can be realized
through Josephson tunneling and Equation (0.2) through both Esaki and
Giaever tunneling. Interestingly, these three young researchers shared the
Nobel Prize in physics in 1973.
Bob’s doctoral research was concerned with both theoretical and exper-
imental studies of these two equations, and his thesis was characterized
by two unique features: it was entirely his own work and it was easily the
shortest thesis that I have ever approved. Looking through The Supercon-
ductive Tunnel Junction Neuristor today, I am impressed by his simple and
direct prose, and filled again with the delicious sense of how exciting was
nonlinear science in those early days. So much was sitting just in front of

us, waiting to be discovered.
This thesis was a tour de force, consisting of five distinct contributions.
• On the theoretical side, he introduced the idea of studying traveling
v
wave stability in a moving frame, using this concept to establish the
stability of step (or level changing) solutions of Equation (0.2).
• Again theoretically, he considered an augmentation of Equation (0.2)
with a realistic description of superconducting surface impedance,
leading to the hitherto unexpected possibility of a pulse-shaped trav-
eling wave. The existence of such a solution is important if the super-
conducting transmission line is to be employed as a neuristor; a fact
recognized in US Patent Number 3,717,773 “Neuristor transmission
line for actively propagating pulses,” which was awarded on February
20, 1973.
• On the experimental side of his research, Bob constructed an elec-
tronic transmission line model of the superconducting neuristor—
using Esaki tunnel diodes—demonstrating that his neuristor does in-
deed have pulse-like solutions. Nowadays, this sort of check would be
done on a digital computer, but in the 1960s electronic modeling was
an effective, if tedious, approach.
• Extending fabrication procedures previously developed in our labora-
tory, he constructed tin–tin oxide–lead superconducting tunnel trans-
mission lines of the Giaever type, showing that they could function as
neuristors by propagating traveling pulses as predicted by his theory.
This part of the research was a major effort, involving the making
of 80 superconducting transmission lines, of which only 8 (all con-
structed during winter months when the air in the laboratory was
very dry) were usable.
• Finally, Bob fabricated several superconducting transmission lines of
the Josephson type—by reducing the thickness of the oxide layer—

and showed that they could support pulse-like solutions of varying
speeds, in agreement with the properties of Equation (0.1). These
were the first such systems ever constructed.
All of this work was clearly presented in 94 double spaced pages—to
which I do not recall making a single editorial correction—leading me to
suspect (only half in jest) that the worth of a thesis is inversely proportional
to its weight.
But it would be incorrect to leave the impression that Bob occupied himself
only with scientific matters, for his social conscience was keenly developed.
As the folly of the Vietnam War unfolded throughout the 1960s and the
city of Madison became polarized into flocks of “hawks” and “doves,” he
was in the vanguard of Americans working for an end to the killing and
a peaceful resolution of the conflict. Although those were difficult years
for the University of Wisconsin, the activities of concerned and committed
students like Bob showed it to be a truly great educational institution.
vi
Having completed his thesis in September of 1967, he spent the 1967–68
academic year as a postdoctoral assistant in the Electronics Department
of Professor Georg Bruun at the Technical University of Denmark, where
a group was then engaged in a substantial program of neuristor research.
It was during this period that Bob took the opportunity to visit Prague
and share the euphoria of that beautiful city in its short-lived release from
foreign domination, an experience that left a strong impression, deepening
his suspicion of the motivations behind many official actions.
In the fall of 1968 Bob was recruited by Wisconsin’s Electrical Engineer-
ing Department as a tenure track assistant professor, a signal honor for
the department then had a firm policy against hiring its own graduates in
order to avoid “inbreeding.” The reasons for this departure from standard
procedure was that integrated circuit technology was becoming an impor-
tant aspect of solid state electronics, and both Jim Nordman and I were

fully occupied with our own research activities. As the most competent
person we knew, Bob was brought on board and charged with developing
an integrated circuits laboratory.
Not surprisingly, he was also caught up by the general feeling of student
unrest that characterized those days, eagerly embracing novel approaches
to teaching that would supersede the dull habits of the past. Following
his lead, we presented some courses together on the relationships between
modern technology and national politics that attracted both graduate and
undergraduate students from a wide spectrum of university departments.
One such class, I recall, met by an evening campfire in a wooded park
on Madison’s Lake Mendota, where we would sit in a circle discussing
philosophy, science, technology, and politics as the twilight deepened. The
circle is important. Under Bob’s inspiration, we were all students—the
highest status of an academic—striving together to understand.
So two salient characteristics of Bob’s nature become evident: a sure-
footed and independent approach to his professional work, and a deeply
rooted concern for the spiritual health of his society. But there was more.
Bob had a way of quietly influencing events, of deftly intervening at the
critical moment without worrying about taking credit for the results. From
Denmark in the spring of 1968, he wrote that I should look at the papers
of one E R Caianiello, who was doing interesting work on the theory of
the brain, a vast subject toward which Bob’s neuristor studies beckoned.
Upon being contacted, Professor Caianiello responded that he would be
pleased to deliver some reprints in person, as he was soon to be visiting in
Chicago. Over a lunch by the lake, I vividly recall, he sketched plans for
the Laboratorio di Cibernetica, a new sort of research institution that was
then being launched in the village of Arco Felice, near Naples.
Following ideas that had been advanced a decade before by the Ameri-
can mathematician Norbert Wiener, the Laboratorio staff would comprise
mathematicians, physicists, engineers, chemists, computer scientists, elec-

trophysiologists, and neurobiologists—working in a collaborative effort to
vii
understand the dynamic nature of a brain. As Wayne Johnson (who was
just completing an experimental doctorate in superconductive devices) and
I marveled at the scope of this scheme, Eduardo paused, looking thought-
fully at Wayne, and said: “I want you to come to Arco Felice and make
Josephson junctions.” In that moment, the Naples–Madison axis began.
Bob was the fourth Madisonian to trek to the Laboratorio, and the expe-
rience took hold of his psyche to an unanticipated degree. Encouraged by
some subtle cultural chords, it seems, this Wisconsin boy felt immediately
at home. There was something in the air of the mezzogiorno that resonated
with deeper aspects of his spirit. Was it the haunting presence of Homer’s
“wine dark sea” or the glow of afternoon sunlight on Vesuvio’s gorse? Or
the exuberant dance of the olive trees in an autumn breeze, their silver
underskirts flashing in the sun? Contributing perhaps to Bob’s sense of be-
longing to Campania was the marvelous cucina napoletana and the fierce
humor and independence of a people who have endured centuries of foreign
domination. All of these reasons and more, I suspect, drew Bob into the
bosom of Southern Italy.
Madison’s loss was the gain of Naples as Bob carried his talent and ex-
perience in integrated circuit technology into this new environment, deftly
wedding the new photo-lithographic fabrication techniques to emerging
studies of nonlinear wave propagation on long Josephson junctions. Through-
out the 1970s, theoretical, numerical and experimental research in the
nonlinear science of Josephson transmission lines—described by physically
motivated perturbations of the sine-Gordon equation—began to grow and
prosper under the leadership of Bob and Antonio Barone and their students
and colleagues, now far too many to list.
Although our personal and professional lives were entwined over more
than three decades, Bob and I published very little together. One excep-

tion, of which I am particularly proud, was a paper that emerged from a
famous soliton workshop that he organized in the summer of 1977 at the
University of Salerno, to which he had moved a couple of years earlier. Held
at the old quarters of the Physics Department in the middle of the city,
this meeting attracted several stars of nonlinear science and provided un-
usual opportunities for real scientific and personal interactions. One formal
talk in the morning was followed by lunch at a local restaurant that would
have pleased Ernest Hemingway, lasting for a minimum of three hours and
boasting unbounded conversation. Then in the late afternoon we would
gather for another formal talk, after which smaller groups would carry on
into the evening. It was from this inspired disorganization—perhaps only
possible in the mezzogiorno—that it became clear how to solve Equation
(0.1) with boundary conditions, making possible the analytic calculation
of zero field steps in long (but finite) Josephson junctions.
In the mid-seventies, Bob’s bent for subtly influencing events was exer-
cised again. Having become friends with Niels Falsig Pedersen through
meetings at international conferences, Bob encouraged the initiation of
viii
studies on Josephson junction solitons among physicists and applied math-
ematicians at DTU, anticipating the advantages that could be gained from
a collaboration between those near the top and bottom (geographically
speaking) of Europe. During the 1980s, as is evident from several chapters
of this book, such research came of age. In the best traditions of non-
linear science, a remarkable m´enage `a trois of experimental, theoretical
and numerical work emerged, relating the deep insights of soliton theory
to a growing spectrum of experimental observations on long Josephson
junctions. Reflecting the earlier Hodgkin-Huxley work on nerves, this in-
ternational effort serves as a paradigm of how nonlinear science should be
conducted.
Throughout these developments, Bob’s steady influence was ever present,

leading the group mind away from the abrasive competition that is all
too common in many areas of modern science. Much of the civilized tone
characterizing current investigations of superconductive devices stems from
Bob’s guiding hand.
Looking wistfully back over these fleeting years, I see a paradox in Bob’s
nature. Although ever tolerant of human foibles, sensitive to cultural im-
peratives, and ready to seek an intelligent compromise among conflicting
personalities, he remained wary to the end of petty bureaucrats and mean
spirited power games. Indeed, the last email messages we exchanged in
December of 1996 were about codes for protecting internet users against
prying officials of government.
Heading into the twenty-first century, practitioners of nonlinear science
will miss Bob’s wise and gentle counsel. While discussing his tragic death,
Antonio Barone mentioned that in such cases, one often remarks that the
departed person was a “good guy.”
“But, you know,” said Antonio, “Bob, he really was a good guy.”
This is page ix
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PREFACE
The world of science has seen many successes over the past century, but
none has been more striking than the recent flowering of nonlinear research.
Largely ignored in the realms of physics until some three decades ago, stud-
ies of the emergence of coherent structures from the underlying nonlinear
dynamics is now a vital facet of applied and theoretical science, providing
ample evidence—for those who still need it—that
The whole is more than the sum of its parts.
In this book, twenty-eight distinguished contributors describe these devel-
opments from the perspectives of their individual interests, paying partic-
ular attention to those aspects that seem to be of importance for the the
coming century. Although the chapters included here comprise but a small

portion of the current activities, we expect the readers to be impressed by
its diversity and challenge.
The story opens with two fundamental chapters, underlying all of the
others. The first of these presents a general description of coherent phenom-
ena in a variety of experimental settings, including plasma physics, fluid
dynamics and nonlinear optics. The second is a review of developments in
perturbation theory that have been profoundly influenced by research in
nonlinear science since the mid 1960s.
The next four chapters describe various studies of Josephson junction su-
perconductive devices, which have both stimulated and been encouraged by
corresponding developments in nonlinear science. Undreamed of 40 years
ago, these devices have increased the sensitivities of magnetometers and
voltmeters by several orders of magnitude, and they promise corresponding
advances in submillimeter wave oscillators and in the speed of digital com-
putations. Not unrelated to recent progress in the development of supercon-
ductors with higher operating temperatures are the quasi-two-dimensional
magnets that support vortex structures as described in Chapter 7. This is an
exciting field of theoretical study that stems directly from recent advances
in condensed matter physics.
Without doubt, the most important technical application of the ubiqui-
tous and hardy soliton is as a carrier of digital information along optical
fibers. In recognition of this, five chapters are included on various aspects
x
of modern optical research, ranging from general studies of basic proper-
ties to more detailed considerations of current design objectives. We believe
these chapters will provide the reader with an unusually clear exposure to
both the theoretical and the practical implications of optical solitons for
the coming century.
Another significant branch of present day nonlinear science is that of non-
linear lattices. Going back to the early 1980s, this work is introducing the

revolutionary concept of local modes into the study of molecular crystals.
Of the six chapters in this area, the first deals with dislocation dynam-
ics in crystals, and the second suggests the key role that two-dimensional
breathers may have played in the formation of crystal structures, such as
muscovite mica. Other chapters deal with novel phenomena arising from
more than one length scale, mechanical models for lattice solitons, and
the quantum theory of solitons in real lattices. From such work, we be-
lieve, may emerge basic elements for coherent information processing in
the terahertz (far infra-red) region of the electromagnetic spectrum. The
final chapter in this nonlinear lattice segment of the book describes ways
in which “colored” thermal noise can give rise to molecular motors at the
scale of nanometers. This idea has important implications for transport
mechanisms that may operate within living cells, setting the stage for the
final four chapters which address the nonlinear science of life.
Just as the past 100 years have been called the “century of physics,” we
expect that the next will be recognized as the “century of biology.” Since
almost every aspect of biology is nonlinear, this is the area in which we see
the new ideas having their greatest impact. Thus the last four chapters are
devoted to physical aspects of biological research.
The first of these describes various attempts to understand the dynamics
of DNA in the context of modern biophysics. This survey provides the
reader with a hierarchy of mathematical models, each gaining in accuracy
as the computational difficulties correspondingly increase. Related to this
chapter is the following one, describing exact numerical solutions for the
dynamics of certain helical biomolecules that are components of natural
protein.
The penultimate chapter—on exploratory investigations of the nonlinear
dynamics of bacterial populations—is intended to draw physical scientists
into the study of life. Similarly, the final chapter attempts to encourage
young experimentalists and theorists to consider the most intricate dy-

namical system in the known universe: the human brain.
It is our hope that the readers of this book will make a significant con-
tribution to research activities in the “century of biology.”
Lyngby and Tucson The Editors
1999
This is page xi
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Contents
I Nonlinear Science 1
1 Nonlinear Coherent Phenomena in Continuous Media 3
E.A. Kuznetsov and V.E. Zakharov
1 Introduction 3
2 Phase randomization in nonlinear media 4
3 Nonlinear Schr¨odinger equation 9
4 Solitons in the focusing NSLE 11
5 Collapses in the NLSE 15
6 Weak, strong and superstrong collapses 18
7 Anisotropic black holes 21
8 Structure in media with weak dispersion 26
9 Singularities on a fluid surface 32
10 Solitons and collapses in the generalized KP equation . . . 34
11 Self-focusing in the boundary layer 38
12 References 42
2 Perturbation Theories for Nonlinear Waves 47
L. Ostrovsky and K. Gorshkov
1 Introduction 47
2 Modulated waves 49
3 Direct perturbation method 51
4 Perturbation theories for solitary waves 52
4.1 Direct perturbation method for solitons:

quasistationary approach 52
4.2 Nonstationary approach 54
4.3 Inverse-scattering perturbation scheme 55
4.4 “Equivalence principle” 57
4.5 Example: soliton interaction in Lagrangian systems . 58
4.6 Radiation from solitons 59
5 Asymptotic reduction of nonlinear wave equations 60
6 Conclusions 61
7 References 62
xii Contents
II Superconductivity and Magnetism 67
3 Josephson Devices 69
A. Barone and S. Pagano
1 Introduction 69
2 Elements of the Josephson effect 70
3 SQUIDs 72
4 Digital devices 75
5 Detectors 77
6 Voltage standard 78
7 Microwave oscillators 80
8 Conclusions 83
9 References 83
4 Josephson Flux-Flow Oscillators in Microwave Fields 87
M. Salerno and M. Samuelsen
1 Introduction 87
2 Flux-flow oscillators in uniform microwave fields 88
3 Flux-flow oscillators in non-uniform microwave fields 91
4 Numerical experiment 94
5 Conclusions 98
6 Appendix 98

7 References 100
5 Coupled Structures of Long Josephson Junctions 103
G. Carapella and G. Costabile
1 Stacks of two long Josephson junctions 103
1.1 The physical system and its model 103
1.2 Experiments on stacks
of two long Josephson junctions 106
2 Parallel arrays of Josephson junctions 109
2.1 The physical system and its model 109
2.2 Numerical and experimental results
on five-junctions parallel arrays 112
3 Triangular cells of long Josephson junctions 113
3.1 Themodel 114
3.2 Numerical and experimental results 115
4 References 118
6 Stacked Josephson Junctions 121
N.F. Pedersen
1 Introduction 121
2 Short summary of fluxon properties 121
3 Stacked junctions 123
4 Fluxon solutions, selected examples 125
Contents xiii
4.1 The coherent 2-fluxon mode 125
4.2 The two modes of the two fluxon case 127
5 Stacked junction plasma oscillation solutions 128
6 Conclusion 135
7 References 135
7 Dynamics of Vortices in Two-Dimensional Magnets 137
F.G. Mertens and A.R. Bishop
1 Introduction 137

2 Collective variable theories at zero temperature 141
2.1 Thiele equation 141
2.2 Vortex mass 143
2.3 Hierarchy of equations of motion 146
2.4 An alternative approach: coupling to magnons . . . 152
3 Effects of thermal noise on vortex dynamics 155
3.1 Equilibrium and non-equilibrium situations 155
3.2 Collective variable theory
and Langevin dynamics simulations 155
3.3 Noise-induced transitions
between opposite polarizations 161
4 Dynamics above the Kosterlitz-Thouless transition 163
4.1 Vortex-gas approach 163
4.2 Comparison with simulations and experiments . . . 164
4.3 Vortex motion in Monte Carlo simulations 165
5 Conclusion 166
6 References 167
III Nonlinear Optics 171
8 Spatial Optical Solitons 173
Yu.S. Kivshar
1 Introduction 173
2 Spatial vs. temporal solitons 175
3 Basic equations 176
4 Stability of solitary waves 178
4.1 One-parameter solitary waves 179
4.2 Two-parameter solitary waves 180
5 Experiments on self-focusing 182
6 Soliton spiralling 184
7 Multi-hump solitons and solitonic gluons 186
8 Discrete spatial optical solitons 189

9 References 191
xiv Contents
9 Nonlinear Fiber Optics 195
G.P. Agrawal
1 Introduction 195
2 Fiber characteristics 196
2.1 Single-mode fibers 196
2.2 Fiber nonlinearities 196
2.3 Group-velocity dispersion 197
3 Pulse propagation in fibers 198
3.1 Nonlinear Schr¨odinger equation 198
3.2 Modulation instability 198
4 Optical solitons 199
4.1 Bright solitons 200
4.2 Dark solitons 201
4.3 Loss-managed solitons 202
4.4 Dispersion-managed solitons 203
5 Nonlinear optical switching 205
5.1 SPM-based optical switching 205
5.2 XPM-based optical switching 207
6 Concluding remarks 209
7 References 209
10 Self-Focusing and Collapse of Light Beams
in Nonlinear Dispersive Media 213
L. Berg´e and J. Juul Rasmussen
1 Introduction 213
2 General properties of self-focusing
with anomalous group velocity dispersion 214
2.1 Basic properties 214
2.2 Self-similar wave collapses 216

3 Self-focusing with normal group velocity dispersion 220
4 Discussion of the general properties, outlook 224
5 References 226
11 Coherent Structures
in Dissipative Nonlinear Optical Systems 229
J.V. Moloney
1 Introduction 229
2 Nonlinear waveguide channeling in air 234
2.1 Dynamic spatial replenishment of femtosecond pulses
propagating in air 236
3 Control of optical turbulence in semiconductor lasers 239
3.1 The control 240
4 Summary and conclusions 243
5 References 245
Contents xv
12 Solitons in Optical Media with Quadratic Nonlinearity 247
B.A. Malomed
1 Introduction 247
2 The basic theoretical models 249
3 The solitons 253
4 Conclusion 260
5 References 261
IV Lattice Dynamics and Applications 263
13 Nonlinear Models for the Dynamics
of Topological Defects in Solids 265
Yu.S. Kivshar, H. Benner and O.M. Braun
1 Introduction 265
2 The FK model and the SG equation 266
3 Physical models 269
3.1 Dislocations in solids 269

3.2 Magnetic chains 270
3.3 Josephson junctions 271
3.4 Hydrogen-bonded chains 273
3.5 Surface physics and adsorbed atomic layers 275
3.6 Models of the DNA dynamics 276
4 Properties of kinks 277
4.1 On-site potential of general form 277
4.2 Discreteness effects 279
4.3 Kinks in external fields 280
4.4 Compacton kinks 281
5 Experimental verifications 281
5.1 Josephson junctions 281
5.2 Magnetic systems 283
6 Concluding remarks 285
7 References 286
14 2-D Breathers and Applications 293
J.L. Mar´ın, J.C. Eilbeck and F.M. Russell
1 Introduction 293
2 Deciphering the lines in mica 294
3 Numerical and analogue studies 296
4 Longitudinal moving breathers in 2D lattices 299
5 Breather collisions 302
6 Conclusions and further applications 303
6.1 Application to sputtering 303
6.2 Application to layered HTSC materials 303
7 References 304
xvi Contents
15 Scale Competition in Nonlinear Schr¨odinger Models 307
Yu. B. Gaididei, P.L. Christiansen and S.F. Mingaleev
1 Introduction 307

2 Excitations in nonlinear Kronig-Penney models 308
3 Discrete NLS models with long-range dispersive interactions 311
4 Stabilization of nonlinear excitations by disorder 316
5 Summary 319
6 References 320
16 Demonstration Systems for Kink-Solitons 323
M. Remoissenet
1 Introduction 323
2 Mechanical chains with double-well potential 325
2.1 Chain with torsion and gravity 325
2.2 Chain with flexion and gravity 329
2.3 Numerical simulations 330
3 Experiments 331
3.1 Chain with torsion and gravity 331
3.2 Chain with flexion and gravity 332
4 Lattice effects 333
5 Conclusion 334
6 Appendix 335
7 References 336
17 Quantum Lattice Solitons 339
A.C. Scott
1 Introduction 339
2 Local modes in the dihalomethanes 339
2.1 Classical analysis 340
2.2 Quantum analysis 341
2.3 Comparison with experiments 344
3 A lattice nonlinear Schr¨odinger equation 344
4 Local modes in crystalline acetanilide 348
5 Conclusions 354
6 References 355

18 Noise in Molecular Systems 357
G.P. Tsironis
1 Introduction 357
2 Additive correlated ratchets 358
3 Current reversal 363
4 Synthetic motor protein motion 364
5 Targeted energy transfer and nonequilibrium fluctuations
in bioenergetics 368
6 References 369
Contents xvii
V Biomolecular Dynamics and Biology 371
19 Nonlinear Dynamics of DNA 373
L.V. Yakushevich
1 Introduction 373
2 General description of DNA dynamics.
Classification of the internal motions 375
3 Mathematical modeling of DNA dynamics.
Hierarchy of the models 376
3.1 Principles of modeling 376
3.2 Structural hierarchy 376
3.3 Dynamical hierarchy 377
4 Nonlinear mathematical models.
Solved and unsolved problems 379
4.1 Ideal models 379
4.2 Nonideal models 380
4.3 Statistics of solitons in DNA 380
5 Nonlinear DNA models and experiment 381
5.1 Hydrogen-tritium (or hydrogen-deuterium) exchange 381
5.2 Resonant microwave absorption 382
5.3 Scattering of neutrons and light 383

5.4 Fluorescence depolarization 384
6 Nonlinear conception and mechanisms of DNA functioning . 385
6.1 Nonlinear mechanism of conformational transitions . 385
6.2 Nonlinear conformational waves and long-range effects385
6.3 Direction of transcription process 386
7 References 387
20 From the FPU Chain to Biomolecular Dynamics 393
A.V. Zolotaryuk, A.V. Savin and P.L. Christiansen
1 Introduction 393
2 Helices in two and three dimensions 394
3 Equations of motion for a helix backbone 397
4 Small-amplitude limit 398
5 Three-component soliton solutions 400
5.1 3D case: solitons of longitudinal compression 402
5.2 2D case: other types of solutions 404
6 Conclusions 405
7 References 407
21 Mutual Dynamics of Swimming Microorganisms
and Their Fluid Habitat 409
J.O. Kessler, G.D. Burnett and K.E. Remick
1 Introduction 409
2 Bioconvection (I) 411
xviii Contents
2.1 Observations 411
2.2 Continuum theory 412
3 Bacteria in constraining environments (II) 415
4 Possibilities for computer simulation 417
5 Conclusion 421
6 Appendix I: Statistical methods 423
7 Appendix II 424

8 References 425
22 Nonlinearities in Biology: The Brain as an Example 427
H. Haken
1 Introduction 427
2 Some salient features of neurons 427
3 The noisy lighthouse model of a neural network 429
4 The special case of two neurons 430
5 Time-averages 433
6 The averaged neural equations 434
7 How to make contact with experimental data?
Synergetics as a guide 440
8 Concluding remarks and outlook 443
9 References 444
Index 447
This is page xix
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List of Authors
Agrawal, Govind P. Barone, Antonio
The Institute of Optics Istituto di Cibernetica
University of Rochester Via Toiano 6
PO Box 270186 I-80072 Arco Felice
Rochester, NY 14627 Napoli
USA Italy
Benner, Hartmut Berg´e, Luc
Institut f¨ur Festk¨orperphysik Commissariat `a l’Energie Atomique
and SFB 185 CEA-DAM/
ˆ
Ile-de-France
Technische Hochschule Darmstadt B.P. 12
Hochschulstrasse 6 F-91680 Bruy`eres-le-Chˆatel

D-64289 Darmstadt France
Germany
Bishop, Alan R. Braun, Oleg M.
Theoretical Division and CNLS Institute of Physics
Los Alamos National Laboratory Ukrainian Academy of Sciences
MS B262 46 Science Avenue
Los Alamos, NM 87545 UA-252022 Kiev
U. S. A. Ukraine
Burnett, G. David Carapella, G.
Physics Department, Building 81 Dipartimento de Fisica
University of Arizona Universit`a di Salerno
Tucson, AZ 85721 I-84081 Baronissi
USA Italy
Christiansen, Peter L. Costabile, Giovanni
Department of Mathematical Modelling Dipartimento de Fisica
Technical University of Denmark Universit`a di Salerno
DK-2800 Lyngby I-84081 Baronissi
Denmark Italy
Eilbeck, J. Chris Gaididei, Yuri B.
Department of Mathematics Institute for Theoretical Physics
Heriot-Watt University Academy of Sciences of Ukraine
Riccarton Metrologicheskaya Street 14-B
Edinburgh EH15 4AS UA-252143 Kiev
United Kingdom Ukraine
xx Contents
Gorshkov, Konstantin A. Haken, Herman
Institute of Applied Physics Institute for Theoretical Physics 1
Russian Acadamy of Sciences Center of Synergetics
46 Ulyanova Street University of Stuttgart
R-603006 Nizhny Novgorod Pfaffenwaldring 57/IV

Russia D-70550 Stuttgart
Germany
Kessler, John O. Kivshar, Yuri S.
Physics Department, Building 81 Optical Sciences Center
University of Arizona The Australian National University
Tucson, AZ 85721 Canberra ACT 0200
USA Australia
Kuznetsov, E. A. Malomed, Boris A.
L.D. Landau Institute Department of
for Theoretical Physics Interdisciplinary Studies
2 Kosygina Street Faculty of Engineering
R-117334 Moscow Tel Aviv University
Russia Tel Aviv 69978
Israel
Mar´ın, J. L. Mertens, Franz G.
Department of Mathematics Physikalisches Institut
Heriot-Watt University Lehrstuhl f¨ur Theoretische Physik I
Riccarton Universit¨at Bayreuth
Edinburgh EH15 4AS Postfach 101251, D-8580 Bayreuth
United Kingdom Germany
Mingaleev, Sergei F. Moloney, Jerry V.
Institute for Theoretical Physics Department of Mathematics
Academy of Sciences of Ukraine and Optical Science
Metrologicheskaya Street 14-B University of Arizona
UA-252143 Kiev Tucson, Arizona 85721
Ukraine USA
Ostrovsky, Lev and
University of Colorado Institute of Applied Physics
Cooperative Institute for Research Russian Acadamy of Sciences
in Environmental Sciences/NOAA 46 Ulyanova Street

Environmental Technology Laboratory R-603006 Nizhny Novgorod
325 Broadway Russia
Boulder, Colorado 80303
USA
Contents xxi
Pagano, Sergio Pedersen, Niels Falsig
Istituto de Cibernetica del CNR Department of Electric
Via Toiano 6 Power Engineering
Arco Felice, NA Napoli Technical University of Denmark
Italy DK-2800 Lyngby
Denmark
Rasmussen, Jens Juul Remick, Katherine E.
Risø National Laboratory Neuroscience
Optics and Fluid Dynamics Department University of Texas Medical Branch
P.O. Box 49 Galveston, TX 77555
DK-4000 Roskilde USA
Denmark
Remoissenet, Michel Russell, F. M.
Facult´e des Sciences et Techniques Department of Mathematics
Universit´e de Bourgogne Heriot-Watt University
9 Av. A. Savary BP400 Riccarton
F-21011 Dijon Edinburgh EH15 4AS
France United Kingdom
Salerno, Mario Samuelsen, Mogens
Dipartimento di Fisica Teorica Department of Physics
Universit`a di Salerno Technical University of Denmark
Via S. Allende DK-2800 Lyngby
I-84081 Baronissi Denmark
Italy
Savin, Alexander V. Scott, Alwyn C.

Institute for Physico-Technical Problems Department of Mathematics
Prechistenka 13/7 University of Arizona
R-119034 Moscow Tucson, AZ 85721
Russia USA
and Tsironis, George P.
Department of Mathematical Modelling Research Center of Crete
Technical University of Denmark and Physics Department
DK-2800 Lyngby University of Crete
Denmark P.O. Box 2208
G-71003 Iraklion
Greece
xxii Contents
Yakushevich, Ludmilla V. Zakharov, V. E.
Institute of Cell Biophysics L.D. Landau Institute for
Academy of Sciences Theoretical Physics
R-142292 Pushchino 2 Kosygina Street
Russia R-117344 Moscow
Russia
Zolotaryuk, Alexander V.
Bogolyubov Institute for
Theoretical Physics
Ukranian Academy of Sciences
Metrologicheskaya Street 14-B
UA-252143 Kiev
Ukraine
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1
Nonlinear Coherent Phenomena in
Continuous Media

E.A. Kuznetsov
V.E. Zakharov
ABSTRACT This review is devoted to description of coherent nonlin-
ear phenomena in almost conservative media with applications to plasma
physics, fluid dynamics and nonlinear optics. The main attention in the
review is paid to consideration of solitons, collapses, and black holes. The
latter is a quasi-stationary singular object which appear after the forma-
tion of a singularity in nonlinear wave systems. We discuss in details the
qualitative reasons of the wave collapse and a difference between solitons
and collapses, and apply to their analysis exact methods based on the
integral estimates and the Hamiltonian formalism. These approaches are
demonstrated mainly on the basic nonlinear models, i.e. on the nonlinear
Schr¨odinger equation and the Kadomtsev-Petviashvili equation and their
generalizations.
1 Introduction
All real continuous media, including vacuum, are nonlinear. Nonlinearity
might be a cause of quite opposite physical effects. One of them is phase
randomization leading to formation of a chaotic state - weak or strong
wave turbulence. Wind-driven waves on the ocean surface is the classical
example of that sort. Another group of effects is spontaneous generation
of coherent structures. These structures may be localized in space or both
in space and in time. Phases of Fourier harmonics, forming the structures,
are strongly correlated.
Very often coherent structures coexist with wave turbulence. A simple
example of the coherent structure is ‘white caps’ on the crest of gravity
wave of high amplitude. Elementary visual observation shows that just
before breaking, a wave crest takes the universal, wedge-type shape. Ap-
parently, the harmonics composing this shape have correlated phases. The
wave breaking is an important mechanism of energy and momentum dis-
sipation on the ocean. A satisfactory theory of this basic effect is not yet

developed.
P.L. Christiansen, M.P. Sørensen, and A.C. Scott (Eds.): LNP 542, pp. 3−45, 2000.
 Springer-Verlag Berlin Heidelberg 2000