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The development of nonlinear science£ Alwyn Scott Department of Mathematics University of Arizona Tucson, Arizona USA August 11, 2005 Abstract Research activities in nonlinear science over the past three centuries are reviewed, paying particular attention to the explosive growth of interest in chaos, solitons, and reaction-diffusion phenomena that occurred during the 1970s and considering whether this explosion was an example of a “sci- entific revolution” or Gestalt-like “paradigm shift” as proposed by Thomas Kuhn in the 1962. The broad structure of modern nonlinear science is sketched and details of developments in several areas of nonlinear research are presented, including cosmology, theories of matter, quantum theory, chemistry and biochemistry, solid-state physics, electronics, optics, hydro- dynamics, geophysics, economics, biophysics and neuroscience. It is con- cluded that the emergence of modern nonlinear science as a collective in- terdisciplinary activity was a Kuhnian paradigm shift which has emerged from diverse areas of science in response to the steady growth of comput- ing power over the past four decades and the accumulation of knowledge about nonlinear methods, which eventually broke through the barriers of balkanization. Implications of these perspectives for twentyfirst-century research in biophysics and in neuroscience are discussed. PACS: 01.65.+g, 01.70.+w, 05.45.-a £ Copyright ­c 2005 by Alwyn C. Scott. All rights reserved. 1 Contents 1 Introduction 6 1.1 What is nonlinear science? ...................... 6 1.2 An explosion of activity . ...................... 10 1.3 What caused the changes? ...................... 12 1.4 Three trigger events .......................... 16 2 Fundamental phenomena of nonlinear science 18 2.1 Chaos theory . ............................. 18 2.2 Solitons and solitary waves ..................... 26 2.3 Reaction-diffusion systems ...................... 40 2.4 Summary . ............................. 48 3 Applications of nonlinear theory 50 3.1 Cosmology . ............................. 50 3.2 Nonlinear theories of matter ..................... 55 3.3 Quantum theory ............................ 61 3.4 Nonlinear chemical and biochemical phenomena . ....... 67 3.5 Condensed-matter physics ...................... 71 3.6 Nonlinear electronics . ...................... 76 3.7 Nonlinear optics ............................ 82 3.8 Nonlinear fluids ............................ 86 3.9 Economic dynamics .......................... 94 3.10 Biophysics . ............................. 96 3.11 Neuroscience ............................. 102 3.12 General perspectives . ...................... 108 4 Conclusions 110 5 Acknowledgments 113 2 List of Figures 1 The annual number of articles in scientific publications that have used the term CHAOS, SOLITON, and REACTION-DIFFUSION in their ti- tles, abstracts, or key words, and the TOTAL of these three plots. (Data from the Science Citation Index Expanded.) The VERHULST curve is calculated from Equation (1) to approximate the TOTAL curve with Æ ¿½¼¼ Æ ´½¼µ ½¾ ¼¾ ¼ , and . (Note that the values of the CHAOS plot are misleadingly high before about 1975, as authors then used the term in its classical meaning.) .............. 10 2 A histogram of the data in Table 1, showing that the number of jour- nals available for publications in nonlinear science increased greatly between 1980 and 2000. ......................... 14 3 The number of transistors on an Intel processor vs. time (data from [256]). .................................. 15 4 Plots of the annual number of citations to Lorenz’s 1963 paper ([311]) and the annual number of papers using the term “chaos”. (Data from Science Citation Index Expanded.) ................... 17 5 Sketch of the Lorenz attractor of a solution trajectory of Equations (3) ½¼ Ö ¾ ¿ with , and . The view is looking down from the ´Ü Þ µ Ý -axis onto the -plane. ....................... 19 6 A bifurcation diagram for solutions of Equation (4). (Courtesy of Mikhail Rabinovich and Nikolai Rulkov.) .................... 22 7 A hydrodynamic soliton created in a wave tank by John Scott Russell in the 1830s. (a) A column of water is accumulated at the left-hand end of the tank. (b) Release of this water by lifting the sliding panel generates a solitary wave that travels to the right with velocity Ú . (Redrawn from [448].) .................................. 27 Ù ´Ü ص 8 A plot of a 2-soliton KdV solution ¾ from Equation (14). ... 31 9 A single soliton (kink) on a mechanical model of Equation (17). .... 33 10 Stroboscopic photograph showing Lorentz contraction of a kink soli- ton on a mechanical model of Equation (17) [460]. (Dissipative effects cause the kink to slow down as it moves to the right, whereupon its size increases.) ................................ 33 11 A stationary breather solution of SG plotted from Equation (22) with . ................................ 34 12 The upper figure shows a Toda lattice at rest. The lower figure shows the Toda-lattice (TL) soliton given in Equation (25). .......... 36 3 13 (a) A computer simulation of the deadly tsunami (an example of Rus- sell’s Great Wave) in the Indian Ocean on 26 December 2004, 1 hour and 50 minutes after it was launched by an earthquake source (green area) near Sumatra. The red colour indicates that the water surface is higher than normal, while the blue colour means that it is lower, and the darker the colour, the larger the amplitude. See http://staff.aist.go.jp/kenji.satake/animation.html for an animated version of this figure. (Courtesy of Kenji Satake, Na- tional Institute of Advanced Industrial Science and Technology, Japan.) (b) Apollo–Soyuz photograph of the Andaman sea, showing surface ev- idence of internal waves [398]. [The location of (b) is shown as a small rectangle in (a).] (Courtesy of Alfred Osborne.) ............ 39 14 An early cathode-ray oscilloscope photograph of a nerve impulse. The dots on the lower edge indicate time in milliseconds, increasing to the right. The solid line shows the time course of the transmembrane volt- age (Î ), becoming increasingly positive inside the axon. The wide band shows the transmembrane permeability which is large (small mem- brane resistance) during the impulse. (Courtesy of Kenneth Cole.) .. 41 15 Diagram showing a squid axon and the measurement of internal volt- age using a glass microelectrode (not to scale). ............. 42 16 In a BZ reaction, two-dimensional waves spiral outward from organiz- ing centers, intersecting to form cusps. (Courtesy of A.T. Winfree.) .. 46 17 A computer generated plot of a scroll ring. (Courtesy of A.T. Winfree.) 47 18 Interrelations among low-dimensional chaotic systems, reaction-diffusion (RD) systems and integrable (solitonic) systems. ............ 49 19 Three-dimensional skyrmions with increasing values of topological charge. (Courtesy of Paul Sutcliffe.) ...................... 59 20 Annual number of articles in Physical Review D that have used the term “soliton” in the title, abstract or key words. (Data from Science Citation Index Expanded.) ............................ 60 21 The planar structure of a benzene molecule, showing a local mode of the CH-stretching oscillation. ...................... 62 ¾ 22 Energy eigenvalues of the DNLS equation at the second (Ò ) quan- ½ ½ tum level with as . There is one soliton band for each ­ value of ­ , which is plotted for several different values of . ..... 65 23 Infrared absorption spectra of crystalline acetanilide (ACN) in the re- gion of the amide-I (CO-stretching) mode as a function of tempera- ½ ture. A normal (delocalized) amide-I band is at 1665 Ñ and a self- ½ localized (soliton) peak is at 1650 Ñ . ................ 70 24 A piece of muscovite mica (overall length is 23 cm). Inside the oval marked “A”, the tracks of several moving breathers appear to be emerg- ing from a charged-particle track. (Courtesy of Mike Russell.) ..... 75 25 A Josephson oscillator in which a quantum of magnetic flux (fluxon) bounces back-and-forth between the two ends of a rectangular junction. 81 26 (a) Sketch of a pump-probe setup. (b) Pump-probe experiment of an excited mode. .............................. 85 4 27 A log-log plot of data from the Trinity explosion on 16 July 1945, show- Ø ing the radius of expansion (Ê) as a function of time ( ) during the first 62 ms. (Data from [505].) ........................ 87 28 A tidal bore (or mascaret) at Quillebeuf, France. The site is about 36 km from the mouth of the Seine, where tides up to 10 m occur. Due to dredging in the 1960s, this striking nonlinear phenomenon has disap- peared. (Courtesy of Jean-Jacques Malandain.) ............ 88 29 Shadowgraph showing shock waves produced by a Winchester 0.308 caliber bullet travelling through air at about 2.5 times the speed of sound. (Courtesy of Ruprecht Nennstiel, Bundeskriminalamt Wiesbaden, Ger- many.) ................................. 89 30 Rogue waves (RWs). (a) A RW approaching the stern of a merchant ship in the Bay of Biscay. (Courtesy of NOAA Photo Library.) (b) A RW produced in the Large Wave Channel in Hannover. (Courtesy of the Coastal Research Centre, Hannover, Germany.) (c) A snapshot at the peak value of a numerical RW computed from a 2 + 1 dimensional NLS system, where the parameters are chosen to correspond to the ocean and axes are labeled in metres. (Courtesy of Alfred Osborne.) ..... 92 31 Natural atmospheric vortices. (a) A dust devil in Death Valley, Cali- fornia. (Courtesy of Dr. Sharon Johnson/GeoImages.) 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