nonlinear science at the dawn of the 21st century

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 en

Lecture Notes in Physics

Editorial Board

R Beig, Wien, Austria

J Ehlers, Potsdam, Germany

U Frisch, Nice, France

K Hepp, Z¨urich, Switzerland

W Hillebrandt, Garching, Germany

D Imboden, Z¨urich, Switzerland

R L Jaffe, Cambridge, MA, USA

R Kippenhahn, G¨ottingen, Germany

R Lipowsky, Golm, Germany

H v L¨ohneysen, Karlsruhe, Germany

I Ojima, Kyoto, Japan

H A Weidenm¨uller, Heidelberg, Germany

J Wess, M¨unchen, Germany

J Zittartz, K¨oln, Germany

The Editorial Policy for Proceedings

The series Lecture Notes in Physics reports new developments in physical research and teaching – quickly, informally, and at a high level The proceedings to be considered for publication in this series should be limited

to only a few areas of research, and these should be closely related to each other The contributions should be

of a high standard and should avoid lengthy redraftings of papers already published or about to be published elsewhere As a whole, the proceedings should aim for a balanced presentation of the theme of the conference including a description of the techniques used and enough motivation for a broad readership It should not

be assumed that the published proceedings must reflect the conference in its entirety (A listing or abstracts

of papers presented at the meeting but not included in the proceedings could be added as an appendix.) When applying for publication in the series Lecture Notes in Physics the volume’s editor(s) should submit sufficient material to enable the series editors and their referees to make a fairly accurate evaluation (e.g a complete list of speakers and titles of papers to be presented and abstracts) If, based on this information, the proceedings are (tentatively) accepted, the volume’s editor(s), whose name(s) will appear on the title pages, should select the papers suitable for publication and have them refereed (as for a journal) when appropriate.

As a rule discussions will not be accepted The series editors and Springer-Verlag will normally not interfere with the detailed editing except in fairly obvious cases or on technical matters.

Final acceptance is expressed by the series editor in charge, in consultation with Springer-Verlag only after receiving the complete manuscript It might help to send a copy of the authors’ manuscripts in advance to the editor in charge to discuss possible revisions with him As a general rule, the series editor will confirm his tentative acceptance if the final manuscript corresponds to the original concept discussed, if the quality of the contribution meets the requirements of the series, and if the final size of the manuscript does not greatly exceed the number of pages originally agreed upon The manuscript should be forwarded to Springer-Verlag shortly after the meeting In cases of extreme delay (more than six months after the conference) the series editors will check once more the timeliness of the papers Therefore, the volume’s editor(s) should establish strict deadlines, or collect the articles during the conference and have them revised on the spot If a delay is unavoidable, one should encourage the authors to update their contributions if appropriate The editors of proceedings are strongly advised to inform contributors about these points at an early stage.

The final manuscript should contain a table of contents and an informative introduction accessible also to readers not particularly familiar with the topic of the conference The contributions should be in English The volume’s editor(s) should check the contributions for the correct use of language At Springer-Verlag only the prefaces will be checked by a copy-editor for language and style Grave linguistic or technical shortcomings may lead to the rejection of contributions by the series editors A conference report should not exceed a total

of 500 pages Keeping the size within this bound should be achieved by a stricter selection of articles and not

by imposing an upper limit to the length of the individual papers Editors receive jointly 30 complimentary copies of their book They are entitled to purchase further copies of their book at a reduced rate As a rule no reprints of individual contributions can be supplied No royalty is paid on Lecture Notes in Physics volumes Commitment to publish is made by letter of interest rather than by signing a formal contract Springer-Verlag secures the copyright for each volume.

The Production Process

The books are hardbound, and the publisher will select quality paper appropriate to the needs of the author(s) Publication time is about ten weeks More than twenty years of experience guarantee authors the best possible service To reach the goal of rapid publication at a low price the technique of photographic reproduction from

a camera-ready manuscript was chosen This process shifts the main responsibility for the technical quality considerably from the p ublisher to the authors We therefore urge all authors and editors of p roceedings to observe very carefully the essentials for the preparation of camera-ready manuscripts, which we will supply on request This applies especially to the quality of figures and halftones submitted for publication In addition,

it might be useful to look at some of the volumes already published As a special service, we offer free of charge L A TEX and TEX macro packages to format the text according to Springer-Verlag’s quality requirements.

We strongly recommend that you make use of this offer, since the result will be a book of considerably improved technical quality To avoid mistakes and time-consuming correspondence during the production period the conference editors should request special instructions from the publisher well before the beginning

of the conference Manuscripts not meeting the technical standard of the series will have to be returned for improvement.

For further information please contact Springer-Verlag, Physics Editorial Department II, Tiergartenstrasse 17, D-69121 Heidelberg, Germany

Series homepage – http://www.springer.de/phys/books/lnpp

P L Christiansen M P Sørensen A C Scott (Eds.)

Nonlinear Science

at the Dawn

of the 21st Century

1 3

P L Christiansen

M P Sørensen

A C Scott

Department of Mathematical Modelling

The Technical University of Denmark

Building 321

2800 Kgs Lyngby, Denmark

Library of Congress Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Nonlinear science at the dawn of the 21st century / P L Christiansen (ed.)

-Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ;

Milan ; Paris ; Singapore ; Tokyo : Springer, 2000

(Lecture notes in p hysics ; 542)

ISBN 3-540-66918-3

ISSN 0075-8450

ISBN 3-540-66918-3 Springer-Verlag Berlin Heidelberg New York

This work is subject to copyright All rights are reserved, whether the whole or part of thematerial is concerned, specifically the rights of translation, reprinting, reuse of illustra-tions, recitation, broadcasting, reproduction on microfilm or in any other way, andstorage in data banks Duplication of this publication or parts thereof is permitted onlyunder the provisions of the German Copyright Law of September 9, 1965, in its currentversion, and permission for use must always be obtained from Springer-Verlag Violationsare liable for prosecution under the German Copyright Law

Springer-Verlag is a company in the specialist publishing group BertelsmannSpringer

© Springer-Verlag Berlin Heidelberg 2000

Printed in Germany

The use of general descriptive names, registered names, trademarks, etc in this publicationdoes not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.Typesetting: Camera-ready by the authors/editors

Cover design: design & production, Heidelberg

Printed on acid-free paper

SPIN:10720717 55/3144/du - 5 4 3 2 1 0

Printer: Opaque thisRemembering 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

current is a periodic function of the magnetic flux From this effect, therelevant nonlinear wave equation for transverse electromagnetic waves on

a strip-line structure takes the form

∂2φ

∂x2 − ∂2φ

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 Translationtransported lumps of water on the Union Canal near Edinburgh

Just as Equation (0.1) can be viewed as a nonlinear augmentation of thestandard wave equation, the system

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 wavesolutions have quite different behaviors Equation (0.1) can be realizedthrough Josephson tunneling and Equation (0.2) through both Esaki andGiaever tunneling Interestingly, these three young researchers shared theNobel Prize in physics in 1973

Bob’s doctoral research was concerned with both theoretical and imental studies of these two equations, and his thesis was characterized

exper-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 wasnonlinear 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

wave stability in a moving frame, using this concept to establish thestability 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 factrecognized in US Patent Number 3,717,773 “Neuristor transmissionline 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 bedone 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 mission lines of the Giaever type, showing that they could function asneuristors by propagating traveling pulses as predicted by his theory.This part of the research was a major effort, involving the making

trans-of 80 superconducting transmission lines, trans-of which only 8 (all structed during winter months when the air in the laboratory wasvery dry) were usable

con-• 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 varyingspeeds, in agreement with the properties of Equation (0.1) Thesewere the first such systems ever constructed

All of this work was clearly presented in 94 double spaced pages—towhich I do not recall making a single editorial correction—leading me tosuspect (only half in jest) that the worth of a thesis is inversely proportional

a peaceful resolution of the conflict Although those were difficult yearsfor the University of Wisconsin, the activities of concerned and committedstudents like Bob showed it to be a truly great educational institution

Having completed his thesis in September of 1967, he spent the 1967–68academic 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 Pragueand share the euphoria of that beautiful city in its short-lived release fromforeign domination, an experience that left a strong impression, deepeninghis suspicion of the motivations behind many official actions

In the fall of 1968 Bob was recruited by Wisconsin’s Electrical ing Department as a tenure track assistant professor, a signal honor forthe department then had a firm policy against hiring its own graduates inorder to avoid “inbreeding.” The reasons for this departure from standardprocedure was that integrated circuit technology was becoming an impor-tant aspect of solid state electronics, and both Jim Nordman and I werefully occupied with our own research activities As the most competentperson we knew, Bob was brought on board and charged with developing

Engineer-an integrated circuits laboratory

Not surprisingly, he was also caught up by the general feeling of studentunrest that characterized those days, eagerly embracing novel approaches

to teaching that would supersede the dull habits of the past Followinghis lead, we presented some courses together on the relationships betweenmodern technology and national politics that attracted both graduate andundergraduate 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 discussingphilosophy, 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 footed and independent approach to his professional work, and a deeplyrooted concern for the spiritual health of his society But there was more.Bob had a way of quietly influencing events, of deftly intervening at thecritical moment without worrying about taking credit for the results FromDenmark in the spring of 1968, he wrote that I should look at the papers

sure-of one E R Caianiello, who was doing interesting work on the theory sure-ofthe brain, a vast subject toward which Bob’s neuristor studies beckoned.Upon being contacted, Professor Caianiello responded that he would bepleased to deliver some reprints in person, as he was soon to be visiting inChicago 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, trophysiologists, and neurobiologists—working in a collaborative effort to

elec-understand the dynamic nature of a brain As Wayne Johnson (who wasjust completing an experimental doctorate in superconductive devices) and

I marveled at the scope of this scheme, Eduardo paused, looking fully at Wayne, and said: “I want you to come to Arco Felice and makeJosephson junctions.” In that moment, the Naples–Madison axis began

thought-Bob was the fourth Madisonian to trek to the Laboratorio, and the

expe-rience took hold of his psyche to an unanticipated degree Encouraged bysome 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? Orthe exuberant dance of the olive trees in an autumn breeze, their silverunderskirts 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 foreigndomination All of these reasons and more, I suspect, drew Bob into thebosom of Southern Italy

Madison’s loss was the gain of Naples as Bob carried his talent and perience in integrated circuit technology into this new environment, deftlywedding the new photo-lithographic fabrication techniques to emergingstudies of nonlinear wave propagation on long Josephson junctions Through-out the 1970s, theoretical, numerical and experimental research in thenonlinear science of Josephson transmission lines—described by physicallymotivated perturbations of the sine-Gordon equation—began to grow andprosper under the leadership of Bob and Antonio Barone and their studentsand colleagues, now far too many to list

ex-Although our personal and professional lives were entwined over morethan three decades, Bob and I published very little together One excep-tion, of which I am particularly proud, was a paper that emerged from afamous soliton workshop that he organized in the summer of 1977 at theUniversity 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 formaltalk in the morning was followed by lunch at a local restaurant that wouldhave pleased Ernest Hemingway, lasting for a minimum of three hours andboasting unbounded conversation Then in the late afternoon we wouldgather for another formal talk, after which smaller groups would carry oninto 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 cised again Having become friends with Niels Falsig Pedersen throughmeetings at international conferences, Bob encouraged the initiation of

exer-studies on Josephson junction solitons among physicists and applied ematicians at DTU, anticipating the advantages that could be gained from

math-a collmath-abormath-ation between those nemath-ar the top math-and bottom (geogrmath-aphicmath-allyspeaking) 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 Josephsonjunctions Reflecting the earlier Hodgkin-Huxley work on nerves, this in-ternational effort serves as a paradigm of how nonlinear science should beconducted

Throughout these developments, Bob’s steady influence was ever present,leading the group mind away from the abrasive competition that is alltoo common in many areas of modern science Much of the civilized tonecharacterizing current investigations of superconductive devices stems fromBob’s guiding hand

Looking wistfully back over these fleeting years, I see a paradox in Bob’snature Although ever tolerant of human foibles, sensitive to cultural im-peratives, and ready to seek an intelligent compromise among conflictingpersonalities, he remained wary to the end of petty bureaucrats and meanspirited power games Indeed, the last email messages we exchanged inDecember of 1996 were about codes for protecting internet users againstprying officials of government

Heading into the twenty-first century, practitioners of nonlinear sciencewill miss Bob’s wise and gentle counsel While discussing his tragic death,Antonio Barone mentioned that in such cases, one often remarks that thedeparted person was a “good guy.”

“But, you know,” said Antonio, “Bob, he really was a good guy.”

Printer: Opaque this

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

of modern optical research, ranging from general studies of basic ties to more detailed considerations of current design objectives We believethese chapters will provide the reader with an unusually clear exposure to

proper-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 frommore than one length scale, mechanical models for lattice solitons, andthe quantum theory of solitons in real lattices From such work, we be-lieve, may emerge basic elements for coherent information processing inthe terahertz (far infra-red) region of the electromagnetic spectrum Thefinal 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 transportmechanisms 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,” weexpect that the next will be recognized as the “century of biology.” Sincealmost every aspect of biology is nonlinear, this is the area in which we seethe new ideas having their greatest impact Thus the last four chapters aredevoted 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 thereader with a hierarchy of mathematical models, each gaining in accuracy

as the computational difficulties correspondingly increase Related to thischapter is the following one, describing exact numerical solutions for thedynamics of certain helical biomolecules that are components of naturalprotein

The penultimate chapter—on exploratory investigations of the nonlineardynamics 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 tribution to research activities in the “century of biology.”

1999

Printer: Opaque this

Contents

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

II Superconductivity and Magnetism 67

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 The model 114

3.2 Numerical and experimental results 115

4 References 118

6Stacked 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

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

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

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

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

16Demonstration 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

V Biomolecular Dynamics and Biology 371

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

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

Printer: Opaque this

List of Authors

Agrawal, Govind P Barone, Antonio

The Institute of Optics Istituto di Cibernetica

University of Rochester Via Toiano 6

Institut f¨ur Festk¨orperphysik Commissariat `a l’Energie Atomique

Technische Hochschule Darmstadt B.P 12

Hochschulstrasse 6 F-91680 Bruy`eres-le-Chˆatel

Germany

Theoretical Division and CNLS Institute of Physics

Los Alamos National Laboratory Ukrainian Academy of Sciences

Los Alamos, NM 87545 UA-252022 Kiev

Physics Department, Building 81 Dipartimento de Fisica

University of Arizona Universit`a di Salerno

Tucson, AZ 85721 I-84081 Baronissi

Christiansen, Peter L Costabile, Giovanni

Department of Mathematical Modelling Dipartimento de Fisica

Technical University of Denmark Universit`a di Salerno

Eilbeck, J Chris Gaididei, Yuri B

Department of Mathematics Institute for Theoretical Physics

Heriot-Watt University Academy of Sciences of Ukraine

Gorshkov, Konstantin A Haken, Herman

Institute of Applied Physics Institute for Theoretical Physics 1Russian Acadamy of Sciences Center of Synergetics

46 Ulyanova Street University of Stuttgart

R-603006 Nizhny Novgorod Pfaffenwaldring 57/IV

Germany

Physics Department, Building 81 Optical Sciences Center

University of Arizona The Australian National University

L.D Landau Institute Department of

for Theoretical Physics Interdisciplinary Studies

2 Kosygina Street Faculty of Engineering

Israel

Department of Mathematics Physikalisches Institut

Heriot-Watt University Lehrstuhl f¨ur Theoretische Physik I

Edinburgh EH15 4AS Postfach 101251, D-8580 Bayreuth

Mingaleev, Sergei F Moloney, Jerry V

Institute for Theoretical Physics Department of MathematicsAcademy of Sciences of Ukraine and Optical Science

Metrologicheskaya Street 14-B University of Arizona

University of Colorado Institute of Applied PhysicsCooperative Institute for Research Russian Acadamy of Sciences

in Environmental Sciences/NOAA 46 Ulyanova Street

Environmental Technology Laboratory R-603006 Nizhny Novgorod

Boulder, Colorado 80303

USA

Pagano, Sergio Pedersen, Niels Falsig

Istituto de Cibernetica del CNRDepartment of Electric

Arco Felice, NA Napoli Technical University of Denmark

DenmarkRasmussen, Jens Juul Remick, Katherine E

Risø National Laboratory Neuroscience

Optics and Fluid Dynamics Department University of Texas Medical Branch

Denmark

Remoissenet, Michel Russell, F M

Facult´e des Sciences et Techniques Department of MathematicsUniversit´e de Bourgogne Heriot-Watt University

9 Av A Savary BP400 Riccarton

Dipartimento di Fisica Teorica Department of Physics

Universit`a di Salerno Technical University of Denmark

Italy

Savin, Alexander V Scott, Alwyn C

Institute for Physico-Technical Problems Department of MathematicsPrechistenka 13/7 University of Arizona

Department of Mathematical Modelling Research Center of Crete

Technical University of Denmark and Physics Department

G-71003 IraklionGreece

Yakushevich, Ludmilla V Zakharov, V E.

Institute of Cell Biophysics L.D Landau Institute forAcademy of Sciences Theoretical PhysicsR-142292 Pushchino 2 Kosygina Street

RussiaZolotaryuk, Alexander V

Bogolyubov Institute for

Printer: Opaque this

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

A more standard example of a coherent structure is a solitary wave on thesurface of shallow water These examples present two major types of coher-ent structures - collapses and solitons Solitons are stationary, spatially lo-calized wave packets, which are very common in nonlinear media Collapsesare almost as wide-spread phenomena as solitons These are catastrophicprocesses of concentration of wave energy in localized space domains lead-ing to absorption of at least part of this energy Collapses are an importantmechanism of the wave energy dissipation in almost conservative media,

in particular, they play essential roles for many methods of fusion plasmaheating

Collapses and solitons are not all the coherent structures that can befound in nonlinear media Rich families of coherent structures exist in ac-tive media, providing the balance between pumping and dissipation Amongthem there are patterns described by the Ginsburg-Landau type equationsand spiral waves in reaction-diffusion systems Rolls and hexagons in theBenard convection are such examples But even in almost conservative me-dia one can find coherent structures different from solitons and collapses.One can mention, for instance, “black holes”, which are persistent localizedregions of the wave energy dissipation arising in some cases after the act

of wave collapse resulting in the formation of a singularity

In this paper we shall discuss coherent structures in almost conservativemedia only We concentrate our attention mostly on collapses and solitons,which are, in our opinion, closely related phenomena In many importantphysical situations, collapse is a result of the soliton instability (for moredetails, see two reviews [1, 2] and references therein) We shall briefly dis-cuss also the theory of black holes in the models describing by the nonlinearSchr¨odinger equation (NLSE) Using the Hamiltonian formalism gives us

an opportunity to study the problem of coherent structures in its maximumgenerality (see also our recent review [3] devoted to this subject) Physicalexamples used in the paper are taken mostly from hydrodynamics, nonlin-ear optics, and plasma physics

2 Phase randomization in nonlinear media

Let as consider wave propagation in a uniform boundless conservativemedium The wave field will be described by the complex normal variable

a k (t), satisfying the equation of motion

∂a k

∂t =−i δH

δa ∗ k

where ω(k) is the dispersion law In this case equation (1.1) is trivially

and, respectively, the trajectory of the system winds on the

infinitely-dimensional torus The phase φ k is defined modulo 2π Therefore for two waves with incommensurable frequencies ω(k) and ω(k1) difference

(or sum) in phases φ k (t) ∓ φ k1(t) = φ k (0)∓ φ k1(0) + (ω(k) ∓ ω(k1))t with time becomes random function on the interval 2π Thus, for continuous dependence ω = ω(k) (except ω(k) =const), the linear dispersion leads to

complete phase randomness for the wave distribution

Now let us introduce into (1.1) a quadratic nonlinearity It is enough toreplace

k1=k2+k3, ω( k1) = ω( k2) + ω( k3) (1.9)

have nontrivial real solutions, as for instance, if ω(0) = 0 and ω  > 0.

Suppose further that at t = 0

where k1, k2, k3 satisfy the equations (1.9) Then at t > 0, in the limit

of small enough intensities of waves, the complex amplitude a k (t) can be

sought in the form

Here the coupling coefficient V = V kk1k2

Equation (1.12) can be easily solved in elliptic functions The initial data

C1= 0, C2= C2(0), C3= C3(0)separate the solution describing growth of C1 In particular, at small time

C1 iV C(0)

2 C3(0)t.

This is the simplest nonlinear process - resonant “mixing” of two matic waves

monochro-The equations (1.12) describe also another very important nonlinear

pro-cess, namely, the decay instability of the monochromatic waves Let at t = 0

C1= Ae iφ , C2= q, C3= iq ∗ e iφ , |q|  A. (1.13)Now for small times

where γ = |v||A| is the growth rate of the so-called decay instability This

solution describes exponential growth of the waves C2, C3 Their phases

(C2=|C2|e iφ2, C3=|C3|e iϕ3) satisfy the condition

Thus, the sum of phases φ2 and φ3 is fixed But a phase of one of the

waves in this pair (phase of q) is quite arbitrary We found that in the

most idealized case (when due to the decay instability only one pair ofmonochromatic waves is excited) this process yields the correlation for sum

of phases of the excited waves and simultaneously introduces to the wave

system an element of randomness, namely, the phase of q In more realistic

case the instability excites a whole ensemble of wave pairs satisfying the

conditions (1.9) up to the accuracy of γ Each exited pair adds one random

phase The exited waves are also unstable Multiplication of the process

of instability has to create in the system a lot of new waves with random

phases and to cause finally complete turbulization of the wave field Wemust stress that this scenario is just a very plausible conjecture It would

be very important to check it by a direct numerical experiment The point ofcommon belief is the following As a result of multiple events of the wavemixing and decay instability, after some time phases become completelyrandom In this case the wave field can be described statistically by thecorrelation function

a k a ∗

k
= n k δ kk
(1.16)

Here n kis the quasi-particle density (or the wave action) This quantity for

sufficiently small wave intensity satisfies the kinetic equation (for detailssee [5])

The kinetic equation accounts for the correlation in wave phases (1.15) in

the first order with respect to the matrix element V kk1k2 that, in ular, provides a nonzero three wave correlation function a k a ∗ k1a ∗

partic-k2 =

J kk1k2δ k−k1−k2 .

The state of the wave field described by the kinetic equation (1.17) iscalled weak turbulence Direct numerical examination of the theory of weakturbulence is one of the most interesting problem in computational physics

H = H0+ H1 and P =

ka k a k ∗ d k. (1.20)

The Equation (1.19) has the additional invariant N =

a k a k ∗ d k If H1

H0the equation (1.19) can be considered as linear and again has a solution(1.4), (1.5), but elementary process of the nonlinear wave interaction arenow different

If wave vectorsk1k2k3 of three monochromatic waves satisfy the tion

condi-−ω(k1) + ω( k2) + ω( k3) = ω( −k1+k2+k3), (1.21)they pump a new wave with the wave vector

This is a “resonant mixing” of wave triads Another type of nonlinear teraction is an instability of monochromatic waves As in the previous casethey lead to excitation of wave pairs In the case of instability of an in-dividual wave with the wave vectork0 there excites a pair with the wavevectorsk2, k3, satisfying the conditions

in-k2+k3= 2k0, ω( k2) + ω( k3) = 2ω( k0). (1.23)

Phases of new waves φ2, φ3are connected with the phase of the initial wave

φ0 by the relation

φ2+ φ3= 2φ0+ π/2. (1.24)

Their difference φ2−φ3is again arbitrary Hence this instability introduces

an element of chaos to the system

Another instability taking place in the system (1.19) is instability ofwave pairs If initially the wave field consists of two monochromatic waveswith wave vectorsk0, k1, two other waves grow exponentially, if their wavevectorsk2, k3satisfy the resonant conditions

k2+k3=k0+k1, ω( k2) + ω( k3) = ω( k0) + ω( k1). (1.25)Now

φ2+ φ3= φ0+ φ1+ π/2.

The phase difference φ2− φ3 is arbitrary again

Combination of instability and wave mixing causes complete zation of phases Weak turbulence in the framework of the model (1.19) isdescribed by the kinetic equation

[n k1n k2n k3+ n k n k2n k3− n k n k1 n k2n k n k1n k3]d k1d k2d k 3.

It should be noted that the equations (1.23) not necessarily have real

solu-tions In an isotropic medium ω = ω( |k|), the sufficient condition for their

existence is ω  > 0; ω  < 0 If ω(0) = 0, ω 

k > 0, ω  > 0, the only solution

of (1.22) is k2=k3=k0 In this case stochastization is less obvious andone has to expect formation of coherent structures We study them in thenext section

3 Nonlinear Schr¨ odinger equation

In some important physical situation, for instance, for waves on the surface

of ideal fluid of finite depth T kk1k2k3 has indeterminacies at k1 =k2 =

function on this submanifold Denote T ( k) = T kkkk Then the equation(1.19) has the exact solution

∂ω

∂k α ∂k β



k=k0 , T = −T (k0).

Going to the frame of reference moving with the group velocity one caneliminate the first space derivative We now obtain

If eigenvalues of the tensor ω αβhave different signs, equation (1.31) yields

instability at any sign of T The domain of the instability in the p-space is

concentrated along the cone

If p  k0, this instability goes to the “second order decay instability”

obeying the resonant conditions (1.23) If all eigenvalues of ω αβ are of thesame sign, instability takes place if

T ω αβ p α p β > 0. (1.34)This instability is called the modulation instability (for details, see [1, 6, 7])

In this case the NLSE can be reduced to the form

iψ t + ∆ψ + 2 |ψ|2ψ = 0. (1.35)

We will call this equation the compact focusing NLSE The domain of theinstability of monochromatic wave is bounded now by the condition

|ω αβ p α p β + β | < |T |A2. (1.36)

If T ω αβ p α p β < 0 , the monochromatic wave is stable and the NLSE can

be simplified to the canonical form

iψ t + ∆ψ − 2|ψ|2ψ = 0. (1.37)This is the compact defocusing NLSE Among non-compact NLSE the mostinteresting ones have the following canonical forms

iψ + ψ − ψ + 2|ψ|2ψ = 0, (1.38)

iψ t+ ∆⊥ ψ − ψ xx+ 2|ψ|2ψ = 0. (1.39)Here ∆⊥ = ∂2/∂y2+ ∂2/∂z2 Equation (1.37) describes nonlinear modula-

tions of gravity waves on a surface of deep water, while (1.38) is applicable

to propagation of electromagnetic wave packets in media with negative(normal) dispersion All species of the NLSE describe some coherent struc-tures Only for the compact cases (1.35) and (1.37) they are studied in aproper degree

4 Solitons in the focusingNSLE

Development of instability of the monochromatic wave (condensate) in theframework of the compact focusing NLSE (1.35) does not lead to formation

of weak-turbulent state directly It leads first to formation of the coherentstructures - solitons or collapses In the quantum mechanical analogy, theNLSE (1.35) describes the motion of a particle in a self-consistent poten-tial with attraction, where the attraction is the main cause of existence ofthe localized coherent structures The nature of these structures depends

essentially on the spatial dimension D The most important coherent

struc-ture in (1.35) has maximal spatial symmetry We will discuss only thesestructures

Equation (1.35) can be rewritten as follows

iψ t + ψ rr+D − 1

r ψ r+ 2|ψ|2ψ = 0, 0 < r < ∞. (1.40)This equation preserves two basic constant of motion: number of particles

where ϕ(r) satisfies the equation

Here ∆ϕ = ϕ rr + (D − 1)ϕ r /r The solution (1.43) is a soliton if ϕ(r) → 0

at r → ∞ and integrals N, X, Y are finite It is possible to show that the

solutions of equation (1.44) for D ≤ 4 decrease exponentially at infinity

and this provides finiteness of the integrals N, X, Y

The solution of equation (1.44) is a stationary point of the Hamiltonian

for fixed number of particles N

The solution of (1.44) can be rescaled: ϕ(r, λ) = λϕ0(λr), where ϕ0(ξ)

satisfies the equation

preserving the number of particles As a result, the Hamiltonian takes a

dependence on the parameter a

H(a) = X

a2 − Y

According to (1.45) at the soliton solution ∂H/∂a | a=1= 0 Using (1.46) it

is easy to get that at these solutions [1, 6]

(1.49), (1.58) make it possible to solve easily the question of soliton stability

If D < 2, H s < 0, and the value a = 1 realizes the minimum of the

Hamiltonian (1.48) Hence in this case one can assume that the soliton isstable This result occurs to be true not only for scaling perturbations butalso for the general ones that can be proved rigorously (see, for instance,[1],[8])

This proof is based on the integral estimates of the Sobolev type Theseinequalities arise as sequences of the general imbedding theorems between

the spaces L p and W1 with the norms,

1/2

.

respectively Namely, there exists such a constant B > 0 so that the

fol-lowing inequality between norms is valid (see, e.g., [9, 10]):

Calculation of the minimum of the r.h.s of this inequality with respect to

scaling parameter α gives the multiplicative variant of the Sobolev

where C is a new constant.

In particular, for p = 4 we have (compare with [11])

This inequality can be improved by finding the best constant C in (1.52).

For this aim consider the functional

J {ψ} = N(4−D)/2 X D/2

so that

To find C consider all extremals of the functional J {ψ} and take among

these the one which gives a minimal value for J Note, this functional is invariant with respect to two independent dilatations: ψ → αψ and r →βr.

Therefore the corresponding Euler-Lagrange equation for the functionalextremum leads to

−ψ + ∇2ψ + 2 |ψ|2ψ = 0,

coinciding with Eq (1.46) for the soliton solutions A minimal value of

J {u} is attained on radically-symmetric distribution without nodes and

simultaneously satisfied by Eq.(1.46) This distribution is the ground statesoliton for the stationary NLSE Hence, with account of (1.49) the bestconstant is equal to

N (4− D)

4− D D

d/2

Here N 0D is the number of particles in the ground state soliton N s

depend-ing on the dimension D For example, in the 1D case N01= 2, at D = 2, according to [14] N02 = 5.84 and for D = 3, N03 = 9.47 [2] As a result,

the inequality (1.52) reads (see, for instance, [1] and [12])

Y ≤ CN(4−d)/2 X d/2

This inequality allows immediately to get a proof of 1D soliton stability

Substituting (1.56) at D = 1 into expression (1.42) for the Hamiltonian and

taking into account relations (1.49 ) we arrive at the following estimate (see,for instance, [13])

H ≥ X − CX 1/2 N 3/2 = H s + (X 1/2 − X 1/2

s )2. (1.57)Thus, a 1D soliton realizes the global minimum (in the given class!) ofthe Hamiltonian and therefore is stable not only with respect to smallperturbations but also against finite ones1

If D > 2, the stationary point yields a positive value of H, so that,

instead of being a minimum in the one-dimensional case, solitons realizethe maximum of the Hamiltonian, which is now unbounded from below and

can take (at a → 0 ) arbitrary large negative values On the other hand,

transformations of the type

inflated without changing its energy and number of particles In the linearapproximation the soliton is marginally stable [15] More detailed studyshows that the soliton is unstable with respect to perturbations of finiteamplitude

The application of the procedure (1.57) at D = 2 gives

H ≥ X 1− N

N02



.

1These inequalities were first used for the stability study of ion-acoustic

soli-tons in magnetized plasma [73] based on the proof of the boundedness of theHamiltonian Later this approach was widely applied for the stability proof ofthe different kinds of solitons (see, for instance, [1]) The acknowledged best use

of these inequalities for the collapse problem was presented by Weinstein [12].Later more general results are reviewed in the paper [2]

From this estimate one can conclude that the Hamiltonian is boundedfrom below, taking non-negative values if the number of particles does not

exceed the number of particles N s at the ground state soliton solution Itsminimal value, equal to zero, is retained for distributions with vanishingmean square value of the wave number,

In the three-dimensional case the analogous integral estimate for H [19],

does not allow us to make any conclusion about soliton stability (note thatmaximum of the r.h.s of (1.59) corresponds to a 3D soliton) Recall thatthe linear stability analysis predicts the instability of three-dimensionalsolitons [1, 15]

For D = 1 equation (1.40) is integrable [16] and has infinite number of

extra constants of motion [16] In this case a soliton can be found in theexplicit form

This is a limiting case for the soliton solutions Now ϕ0vanishes powerfully

at r → ∞, that results in the logarithmical divergence of N0 at r → ∞.

5 Collapses in the NLSE

For D ≥ 2 solitons are either unstable or do not exist In this case the major

coherent structure is a collapsing cavity (the region of higher wave intensity)leading to the formation of localized singularities of wave amplitude in afinite time

One of the main reasons for the wave collapse existence is the nian unboundedness In such systems, like the NLSE, collapse can be rep-resented as a process of falling down of some “particle” in a self-consistent

Hamilto-unbounded potential Indeed, the picture is more complicated than ered above From the very beginning we have a spatially-distributed systemwith infinite number of degrees of freedom and therefore, rigorously speak-ing, it is hard to describe such a system by its reduction to a system ofODEs The NLSE is a wave system and wave radiation plays a very essen-tial role for blow-up.

consid-Let Ω be an arbitrary region with a negative Hamiltonian HΩ Then

using the mean value theorem for the integral YΩ,

Ω =3, it follows that max |ψ|2 as a function of t always is majorized by

the conservative value So, vanishing or yet some sufficient decreasing ofthe initially existed maximum of|ψ|2are impossible

Let the Hamiltonian be negative initially in some separate region Ω,

HΩ < 0, and the radiation emerge from this region In the outer region,

far from Ω, radiative waves will have small amplitudes Consequently, their

nonlinear interaction will be negligible with respect to their dispersion andthey will have a positive Hamiltonian Therefore due to the wave radiation,

the Hamiltonian of the region HΩwill become more and more negative

in-creasing its absolute value, that is possible only due to the unboundedness

of the Hamiltonian Simultaneously, NΩas a positive value will decrease sothat the ratio in the r.h.s of the inequality (1.63) will increase It automat-ically leads to the growth of the maximal value of|ψ|2 Thus, radiation, as

a dissipative process promotes the wave collapse

The occurrence of wave collapses can be proved by use of the virialtheorem From (1.40) one can derive the relation

where the constants C1, C2are defined from the initial conditions Hence it

is seen that for H < 0, in spite of the values C 1,2there always exists a finite

time when the right hand side of (1.65) vanishes Thus, H < 0 represent a

sufficient condition for the collapse, which was found by Vlasov, Petrishchev

and Talanov (the VPT criterion [17]) For D = 3 the equality (1.65) can

be replaced by the inequality

r2 < 4 H

N t

from which follows the same sufficient criterion H < 0 [6] This estimate,

however, is rather rough and can be improved As was shown in a recentpaper [19], the collapse threshold is defined by the unstable ground statesoliton solution which in some sense plays the role of separatrix between

collapsing and noncollapsing solution It was proved in [19] that at D = 3

the equality (1.64) can be changed to the inequality

d2

dt2

r2|ψ|2dr < 8(H − H N ). (1.67)

Here H N = N2/N is the value of the Hamiltonian of the ground state

soliton (compare with (1.49)) Hence the equation (1.67) gives the sharpercriterion for collapse [18, 19]

Here κ = κ(D) is the eigenvalue of the nonlinear boundary problem (1.70).

It is easy to show that as ξ → ∞

A generic case D > 2 can be called supercritical The case D = 2 is

critical This case is especially interesting because it describes stationaryself-focusing of electromagnetic waves in a nonlinear Kerr dielectric

For D ≤ 2 the singularity (1.71) is non-integrable, and the boundary

problem (1.70) cannot have regular solutions In the critical case D =

2, N s = N0, H s= 0, and one can guess that the collapse is the compressingsoliton [24]

In the strictly self-similar case f (ξ) = √

ξ As far the divergence at D = 2

is very weak (logarithmic) one can conjecture that now

f (ξ) =



ξ b(ξ) .

Here b(ξ) is a “slow” function and b(0) = ∞ It was shown [27, 28] that

particles, but, in contrast, along the x-axis repulsion Moreover, from the

virial identities for mean transverse size and mean longitudinal size one canshow that collapse of the wave packet as a whole is impossible at the stage

of the compression of the wave packet in all directions ([35]) Numericalintegration of these equations (as it was published in the first paper [36],devoted to this subject, as well as in the recent one [37]), demonstratesthe fractal behavior of the system The initial distribution with sufficientlylarge amplitude at the beginning demonstrates compression in the trans-verse plane; at the later stage the wave packet undergoes waving instabilitythat results in splitting of the packet into two packets At the next stagedynamics of each secondary packet repeats the fate of the original one

6 Weak, strongand superstrongcollapses

The central problem of the physical theory of collapse is the estimate ofthe efficiency of collapse as a nonlinear mechanism of wave energy dissipa-tion To achieve that we must include the nonlinear dissipative terms into

6 Weak, strongand superstrongcollapses

The central problem of the physical theory of collapse is the estimate ofthe efficiency of collapse as a nonlinear. .. take (at a → ) arbitrary large negative values On the other hand,

transformations of the type

inflated without changing its energy and number of particles In the linearapproximation