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Typesetting: Camera-ready by the authors/editors Cover design: design & production,Heidelberg Printed on acid-free paper SPIN:10720717 55/3144/du-543210 This is page iii Printer: Opaque this 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.” [...]... for the coming century Another significant branch of present day nonlinear science is that of nonlinear 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 dynamics in crystals, and the second suggests the key role that two-dimensional breathers... 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... spatial size (width) of the soliton Independence of λ of the basic constants of motion for D = 2 means that the soliton is “soft” - it can be compressed or inflated without changing its energy and number of particles In the linear approximation the soliton is marginally stable [15] More detailed study shows that the soliton is unstable with respect to perturbations of finite amplitude The application of. .. 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,”... instability of the monochromatic wave (condensate) in the framework of the compact focusing NLSE (1.35) does not lead to formation of weak-turbulent state directly It leads first to formation of the coherent structures - solitons or collapses In the quantum mechanical analogy, the NLSE (1.35) describes the motion of a particle in a self-consistent potential with attraction, where the attraction is the main... 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... application of the procedure (1.57) at D = 2 gives H ≥X 1− 1 N N02 These inequalities were first used for the stability study of ion-acoustic solitons in magnetized plasma [73] based on the proof of the boundedness of the Hamiltonian Later this approach was widely applied for the stability proof of the different kinds of solitons (see, for instance, [1]) The acknowledged best use of these inequalities for the. .. challenge The story opens with two fundamental chapters, underlying all of the others The first of these presents a general description of coherent phenomena 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. .. after the formation 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¨dinger equation and the. .. )−ω(k 2 ) The kinetic equation accounts for the correlation in wave phases (1.15) in the first order with respect to the matrix element V kk 1 k 2 that, in particular, provides a nonzero three wave correlation function ak a∗ k 1 a∗ k 2 = J kk 1 k 2 δ k−k 1 −k 2 The state of the wave field described by the kinetic equation (1.17) is called weak turbulence Direct numerical examination of the theory of weak . (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. Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the. 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.