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experimental mathematics uses the computer to generate qualitative insight where none has existed before. As the visionary of this development, John von Neumann, wrote (in a 1946 article called "On the principles of large scale computing machines"):
"Our present analytic methods seem unsuitable for the solution of the important problems rising in connection with nonlinear partial differential equations and, in fact, with virtually all types of problems in pure mathematics. ... really efficient high-speed computing devices may, in the field of nonlinear partial differential equations as well as in many other fields which are now difficult or entirely denied of access, provide us with those heuristic hints which are needed in all parts of mathematics for genuine progress."
Stan Ulam, together with many of his Los Alamos colleagues, was one of the very first to make this vision a reality. Among Stan's pioneering experimental mathematical investigations was the seminal study of the FPU problem mentioned above. Another example was his early numerical work on nonlinear mappings, carried out in collaboration with Paul Stein (see "Iteration of Maps, Strange Attractors, and Number Theory-An Ulamian Potpourri"). Both of these studies will figure in our later discussion.
The second crucial development has been the experimental observation of "universal" nonlinear characteristics in natural systems that range from chicken hearts and chemical reactors to fluids and plasmas. In the past decade these experiments have reached previously inaccessible levels of precision, so that one can measure quantitative similarities in, for example, the route to chaotic behavior among an enormous variety of nonlinear systems.
The third and final development has been in the area of novel analytical mathematical methods. For instance, the invention of the inverse spectral transform has led to a systematic method for the explicit solution of a large number of nonlinear partial differential equations. Similarly, new methods based on the theory of Hamiltonian systems allow the analysis of nonlinear stability of a wide range of physically relevant mathematical models.
As we shall shortly see, the methodology based on these three developments has been remarkably successful in solving many nonlinear problems long considered intractable. Moreover, the common characteristics of nonlinear phenomena in very distinct fields has allowed progress in one discipline to transfer rapidly to others and confirms the inherently interdisciplinary nature of nonlinear science. Despite this progress, however, we do not have an entirely systematic approach to nonlinear problems. For the general nonlinear equation there is simply no analog of a Fourier transform. We do, however, have an increasing number of well-defined paradigms that both reflect typical qualitative features and permit quantitative analysis of a wide range of nonlinear systems. In the ensuing three sections I will focus on three such paradigms: coherent structures and solitons, deterministic chaos and fractals, and complex configurations and patterns. Of these the first two are well developed and amply exemplified, whereas the third is still emerging. Appropriately, these paradigms reflect different aspects of nonlinearity: coherent structures reveal a surprising orderliness, deterministic chaos illustrates an exquisite disorder, and complex configurations represent the titanic struggle between opposing aspects of order and chaos.
If we were to follow the biblical sequence we would start with chaos, but because it is frankly a rather counterintuitive concept, we shall start with solitons or, more generally and accurately, coherent structures.
Coherent Structures and Solitons
From the Red Spot of Jupiter through clumps of electromagnetic radiation in turbulent plasmas to microscopic charge-density waves on the atomic scale, spatially localized, long-lived, wave-like excitations abound in nonlinear systems. These nonlinear
from earth in the late seventeenth century, has remained remarkably stable in the turbulent cauldron of Jupiter's atmosphere. It represents a coherent structure on a scale of about 4 x 108 meters, or roughly the distance from the earth to the moon.
To give an example at the terrestrial level, certain classes of nonlinear ocean waves form coherent structures that propagate essentially unchanged for thousands of miles. Figure 3b is a photograph taken from an Apollo-Soyuz spacecraft of a region of open ocean in the Andaman Sea near northern Sumatra. One sees clearly a packet of five nearly straight surface waves; each is approximately 150 kilometers wide, so the scale of this phenomenon is roughly 105 meters. Individual waves within the
Fig. 3. (a) A closeup of the Red Spot of Jupiter, taken from the Voyager spacecraft. False color is used to enhance features of the image. In addition to the celebrated Red Spot, there are many other "coherent structures" on smaller scales on Jupiter. (Photo courtesy of NASA). (b) Nonlinear surface waves in the Andaman Sea off the coast of Thailand as photographed from an Apollo-Soyuz spacecraft. (Photo courtesy of NASA.)
Fig. 4. (a) An image, made by using tunnelingelectron microscopy, of a cleaved surface of tantalum diselenide that shows the expected graininess around atomic sites in the crystal lattice. (b) A similar image of tantalum disulfide, showing coherent structures called charge-density waves that are not simply a reflection of the crystal lattice but arise from nonlinear interaction effects. (Photos courtesy of C. G. Slough, W. W. McNairy, R. V. Coleman, B. Drake, and P. K. Hansma, University of Virginia.)
packet are separated from each other by about 10 kilometers. The waves, which are generated by tidal forces, move in the direction perpendicular to their crests at a speed of about 2 meters per second. Although the surface deflection of these waves is smallabout 1.8 meters-they can here be seen from orbit because the sun is directly behind the spacecraft, causing the specular reflection to be very sensitive to variations of the surface. These visible surface waves are actually a manifestation of much larger amplitude-perhaps ten times larger-internal waves. The internal waves exist because thermal or salinity gradients lead to a stratification of the subsurface into layers. A priori such large internal waves could pose a threat to submarines and to off-shore structures. Indeed, the research on these waves was initiated by Exxon Corporation to assess the actual risks to the oil rigs they planned to construct in the area. Fortunately, in this context the phenomenon turned out to be more beautiful than threatening.
Our final physical illustration is drawn from solid-state physics, where the phenomenon of charge-density waves exemplifies coherent structures on the atomic scale. If one studies a crystal of tantalum diselenide using an imaging process called tunnelingelectron microscopy (Fig. 4a), one finds an image that is slightly denser around the atomic sites but otherwise is uniform. Given that the experimental technique focuses on specfic electronic levels, this graininess is precisely what one would expect at the atomic level; there are no nonlinear coherent structures, no charge-density waves. In contrast, tantalum disulfide, which has nearly identical lattice parameters, exhibits much larger structures in the corresponding image (Fig. 4b); in fact, the image shows a hexagonal array of coherent structures. These charge-density waves are separated by about 3.5 normal lattice spacings, so their occurence is not simply a reflection of the natural atomic graininess. Rather, these coherent structures arise because of a nonlinear coupling between the electrons and the atomic nuclei in the lattice. Notice that now the scale is 10-9 meter.
Solitons. We have thus identified nonlinear coherent structures in nature on scales ranging from 108 meters to 10-9 meter-seventeen orders of magnitude! Clearly this paradigm is an essential part of nonlinear science. It is therefore very gratifying that during the past twenty years we have seen a veritable revolution in the understanding of coherent structures. The crucial event that brought on this revolution was the discovery, by Norman Zabusky and Martin Kruskal in 1965, of the remarkable soliton. In a sense, solitons represent the purest form of the coherent-structure paradigm and thus are a natural place to begin our detailed analysis. Further, the history of this discovery shows the intricate interweaving of the various threads of Stan Ulam's legacy to nonlinear science.
To define a soliton precisely, we consider the motion of a wave described by an equation that, in general, will be nonlinear. A traveling wave solution to such an equation is one that depends on the space x and time t variables only through the combination έ = x-vt, where v is the constant velocity of the wave. The traveling wave moves through space without changing its shape and in particular without spreading out or dispersing. If the traveling wave is a localized single pulse, it is called a solitary wave. A soliton is a solitary wave with the crucial additional property that it preserves its form exactly when it interacts with other solitary waves.
The study that led Kruskal and Zabusky to the soliton had its origin in the famous FPU problem, indeed in precisely the form shown in Eq. 6. Experimental mathematical studies of those equations showed, instead of the equipartition of energy expected on general grounds from statistical mechanics, a puzzling series of recurrences of the initial state (see "The Ergodic Hypothesis: A Complicated Problem of Mathematics and Physics"). Through a series of asymptotic approximations, Kruskal and Zabusky related the recurrence question for the system of oscillators in the FPU problem to the nonlinear partial differential equation
Equation 10, called the Korteweg-de Vries or KdV equation, had first been derived in 1895 as an approximate description of water waves moving in a shallow, narrow channel. Indeed, the surface waves in the Andaman Sea, which move essentially in one direction and therefore can be modeled by an equation having only one spatial variable, are described quite accurately by Eq. 10. That this same equation should also appear as a limiting case in the study of a discrete lattice of nonlinear oscillators is an illustration of the generic nature of nonlinear phenomena.
To look analytically for a coherent structure in Eq. 10, one seeks a localized solution us() that depends only on = x vt, thereby reducing the partial differential equation to an ordinary differential equation in έ. The result can be integrated explicitly and, for solutions that vanish at infinity, yields
This solution describes a solitary wave moving with constant velocity v. Moreover, the amplitude of the wave is proportional to v, and its width is inversely proportional to V. The faster the wave goes, the narrower it gets. This relation between the shape and velocity of the wave reflects the nonlinearity of the KdV equation.
Intuitively, we can understand the existence of this solitary wave as a result of a delicate balance in the KdV equation between the linear dispersive term, which tends to cause an initially localized pulse to spread out and change shape as it moves, and the nonlinear convective derivative term u, which tends to increase the pulse where it is already large and hence to bunch up the disturbance. (For a more precise technical analysis of these competing effects in another important nonlinear equation, see "Solitons in the Sine-Gordon Equation.")
Although the solution represented by Eq. 11 is, by inspection, a coherent structure, is it a soliton? In other words, does it preserve its form when it collides with another solitary wave? Since the analytic methods of the 1960s could not answer this question, Zabusky and Kruskal followed another of Ulam's leads and adopted an experimental mathematics approach by performing computer simulations of the collision of two solitary waves with different velocities. Their expectation was that the nonlinear nature of the interaction would break up the waves, causing them to change their properties dramatically and perhaps to disappear entirely. When the computer gave the startling result that the coherent structures emerged from the interaction unaffected in shape, amplitude, and velocity, Zabusky and Kruskal coined the term "soliton," a name reflecting the particle-like attributes of this nonlinear wave and patterned after the names physicists traditionally give to atomic and subatomic particles.
In the years since 1965 research has revealed the existence of solitons in a host of other nonlinear equations, primarily but not exclusively in one spatial dimension. Significantly, the insights gained from the early experimental mathematical studies have had profound impact on many areas of more conventional mathematics, including infinite-dimensional analysis, algebraic geometry, partial differential equations, and dynamical systems theory. To be more specific, the results of Kruskal and Zabusky led directly to the invention of a novel analytic method, now known as the "inverse spectral transform," that permits the explicit and systematic solution of soliton-bearing equations by a series of effectively linear operations. Further, viewed as nonlinear dynamical systems, the soliton equations have been shown to correspond to infinitedegree-of-freedom Hamiltonian systems that possess an infinite number of independent conservation laws and are thus completely integrable. Indeed, the invariance of solitons under interactions can be understood as a consequence of these conservation laws.
Applied Solitons. From all perspectives nonlinear partial differential equations containing solitons are quite special. Nonetheless, as our examples suggest, there is a surprising mathematical diversity to these equations. This diversity is reflected in the
corresponding variety of real-world applications to problems in the natural sciences and engineering. In fiber optics, conducting polymers and other quasi-one-dimensional systems, Josephson transmission lines, and plasma cavitons-as well as the surface waves in the Andaman Sea!-the prevailing mathematical models are slight modifications of soliton equations. There now exist several numerical and analytic perturbation techniques for studying these "nearly" soliton equations, and one can use these to describe quite accurately the behavior of real physical systems.
One specific, decidedly practical illustration of the application of solitons concerns effective long-distance communication by means of optical fibers. Low-intensity light pulses in optical fibers propagate linearly but dispersively (as described in "Solitons in the Sine-Gordon Equation"). This dispersion tends to degrade the signal, and, as a consequence, expensive "repeaters" must be added to the fiber at regular intervals to reconstruct the pulse.
However, if the intensity of the light transmitted through the fiber is substantially increased, the propagation becomes nonlinear and solitary wave pulses are formed. In fact, these solitary waves are very well described by the solitons of the "nonlinear Schrödinger equation," another of the celebrated completely integrable nonlinear partial differential equations. In terms of the (complex) electric field amplitude E(x, t), this equation can be written