Solitary Waves And Solitons

Type of physical science: Solitary Waves and Solitons, Wave-particle duality, Elementary particles, Particles, elementary, Classical physics

Field of study: Electromagnetism

A wave that propagates without dispersion is a solitary wave. If the solitary wave also emerges from an interaction with another solitary wave without modification, the solitary wave is referred to as a "soliton." Solitary waves are found in many fluids, including plasmas, as well as in electromagnetic-wave propagation in optical fibers and other media.

Overview

A wave is a disturbance that propagates from one point to another without any net motion of the medium through which the wave passes. In the case of electromagnetic waves, the notion of a medium must be generalized to include the electric and magnetic fields themselves. Waves therefore transfer energy, but without the net transport of any material. For example, when a sound wave propagates through still air, the disturbance passes through the air transferring sound energy from the source to the receiver, but with no net motion of the air.

A solitary wave is an isolated single peak or trough that propagates without change of shape or speed. Solitary waves are not attenuated as they propagate, nor do they spread. This is in contrast to the more common periodic wave, which consists of a continuous series of peaks and troughs. A special case of the solitary wave is a soliton. Two solitons that collide with each other emerge from the collision without any change of shape. (Except where the distinction is important, the term "solitary wave" will be understood to include solitons for the purposes of this discussion.)

A periodic wave is characterized by a number of parameters, including amplitude, wavelength, frequency, and propagation speed. The behavior of a wave in space and time is governed by a wave equation. In the simplest case of a nondispersive medium, the fundamental solutions of the wave equation are sinusoidal waves, with the waves having the shape of the graph of the trigonometric sine function. Familiar sinusoidal waves include monochromatic (single-wavelength) light, as well as the pure musical tone produced, for example, by a tuning fork. Under ideal conditions, sinusoidal waves would propagate indefinitely without change of shape, amplitude, or frequency. In actual situations, dissipative effects such as friction and fluid viscosity cause the wave to lose amplitude, and other interactions between the wave and the medium can cause alterations of the frequency and speed of the wave.

Even in the absence of these nonideal effects, most waves undergo changes of shape and amplitude as they propagate. This is because most real waves are not purely sinusoidal. Examples include water waves, sound waves produced by most musical instruments or the human voice, and light waves from sources that are not monochromatic. Such nonsinusoidal waves are referred to as "complex waves." Because the shape of such a wave typically changes, the wave equations are more complicated, since they must incorporate the dependence of the wave shape on position and time, in addition to the way the wave as a whole moves through space and time. Fortunately, the mathematical analysis of complex waves is simplified by two remarkable features.

The first of these properties was shown by Thomas Young, an Englishman credited with first having established the wave nature of light. In a linear medium, two or more waves obey the principle of superposition. This principle states that when two or more waves meet at a point, the net disturbance is the algebraic sum of the individual wave disturbances. For example, when the crests of two identical waves coincide, resulting in a net disturbance twice that of either individual wave, "constructive interference" is said to occur. Similarly, if a crest and a trough of two identical waves meet, the result is complete momentary cancellation, or "destructive interference." The superposition of two nonidentical waves leads to a more complicated result. Clarification of the second important feature of wave motion is attributed to the French mathematician Joseph Fourier, who demonstrated that any periodic waveform can be constructed by the superposition of a series of sinusoidal waves of appropriate amplitudes and frequencies. In most cases, the series requires an infinite number of terms to reproduce the periodic waveform exactly, but a finite number of terms will usually give a sufficient approximation to the wave. Fourier further showed that single pulse-type waves can be composed from a continuum, rather than a discrete series, of sinusoidal waves. The result of these two findings is that a complex wave is equivalent to the superposition of a series (or continuum) of sinusoidal waves. Thus, complex waves can be analyzed by looking at the relatively simpler behavior of the sinusoidal waves of which they are composed.

The phenomenon of dispersion can now be understood. The sinusoidal waves of various frequencies that compose a complex wave each travel at a slightly different speed in most media. Generally, higher-frequency waves travel more slowly than lower-frequency waves. As a result, the components of higher frequency begin to lag behind those of lower frequency, and the peak of a wave pulse that starts out tall and narrow is gradually broadened and reduced in amplitude. If there are no dissipative effects, the total wave energy is constant but gradually spread over a larger region, so that the wave amplitude is eventually too small to be detected. Thus, dispersion can destroy a wave even in the ideal case where there is no attenuation caused by friction or other dissipative processes. The most familiar example of dispersion is the spreading of white light into all the component colors of the spectrum by a prism or a raindrop. Dispersion is a general property of all complex waves.

The origin of dispersion can be seen qualitatively in the following way. In order for waves to propagate through some medium, the various parts of the medium must be able to interact with one another so that energy can be transferred. Thus, there is necessarily some form of coupling between the various parts of the medium. The response of the medium to displacements from equilibrium depends on the strength of that coupling and is therefore a function of the frequency of the displacements. Thus, the speed with which the medium can respond to a disturbance and pass the energy along to a neighboring region of the medium is dependent on the frequency of the disturbance. Rarely is wave propagation nondispersive, with the speed of a wave independent of frequency. When the wave speed decreases with increasing frequency, the dispersion is termed "normal." This is the predominant situation for the propagation of light waves in most media. If the wave speed increases with increasing frequency, the dispersion is termed "anomalous."

Since solitary waves are not periodic, they are necessarily complex. In a periodic wave, knowledge of the waveform over one wavelength is sufficient to determine the entire wave everywhere and at all times. In a solitary wave, the displacement is a function of position and time, but it is not a periodic function. Therefore, the functions describing the wave must be known at all points in space and for all times. As a result of Fourier's work, it follows that a solitary wave must be a complex wave, since it requires a continuum of sinusoidal waves spanning all frequencies.

Solitary waves must therefore be subject to dispersion. By definition, however, solitary waves do not show the effects of dispersion; they do not spread out and decrease in amplitude but rather maintain their waveform unchanged. It follows that for solitary waves, there must be some mechanism that compensates for the effects of dispersion. The compensating mechanism results from nonlinearities in the wave propagation. In situations where solitary waves exist in media that exhibit normal dispersion, there is a coupling between the amplitude of a wave and its speed, such that waves of greater amplitude travel faster, and vice versa for anomalous dispersion. To see why this nonlinear behavior stabilizes a solitary wave, consider the case of normal dispersion. As the higher-frequency components begin to lag behind those of lower frequency, the effect is to decrease the amplitude of the solitary wave. The smaller-amplitude component sinusoidal waves travel slower, however, so that the peak begins to overtake the lower-amplitude leading edge of the solitary wave. The wave is steepened--which, in Fourier terms, is equivalent to adding higher-frequency components. Thus, dispersion effectively removes high-frequency components from the solitary wave, while the nonlinear amplitude-speed interaction effectively adds high-frequency components. Equivalently, the solitary wave is flattened and spread by dispersion, while it is steepened and narrowed by nonlinearities. These two opposing effects quickly cause the waveform of the solitary wave to assume a stable shape. It is important to note that the balance that exists between these two opposing effects in a solitary wave is not merely coincidental. The spreading of the wave caused by the dispersion and the steepening effect of the nonlinearity actually adjust the shape of the solitary wave until this balance is achieved. It is generally the case that the properties of the medium in which a wave propagates determine the wave speed. In the simplest case, the wave equations that describe the propagation of the wave are linear equations. It is the linearity of the wave equations that gives rise to the superposition principle, and in such a case, all waves of a given frequency travel at the same speed, regardless of their amplitude. However, in more realistic situations, the wave equations are nonlinear. The properties of the medium determine the wave speed, but the amplitude of the passing wave alters the properties of the medium, which in turn alters the wave speed. In such a system, the superposition principle does not hold. This sort of nonlinear effect is quite common, and even systems that behave linearly for small-amplitude wave propagation will generally become nonlinear for large-amplitude waves. One familiar example of a nonlinear wave system is visible in shallow water, where the nonlinearity causes the waves to "break." A number of nonlinear effects also occur in optical systems, including the "lasing" action of a laser and the optical Kerr effect.

Applications

Solitary waves and solitons are of practical importance for two reasons. First, they occur naturally in a number of physical systems, and understanding these systems at a fundamental level requires an understanding of the role that solitary waves play. Second, they can be used for communications purposes, and potentially for data-storage and computing purposes. As understanding of solitary waves increases, yet more designed applications may be found. The number of situations in which solitary waves are found to play a role is continually increasing. In part, the ever-increasing sophistication of technology has revealed the existence of solitary waves in systems where they were previously unknown.

The first detection of solitary waves in the nineteenth century involved long waves in the shallow water of a canal. Under the proper conditions, similar solitary waves can be seen traveling upstream in rivers that have gently sloping bottoms and wide mouths opening into the ocean, when high tides come ashore. Such solitary waves are known as "tidal bores." One of the better examples of this phenomenon occurs in the Amazon River, where bores of up to five meters in height move upstream at more than twenty kilometers per hour. In China's Chientang River, bores of up to eight meters have been observed. In the several rivers that empty into the Bay of Fundy in Nova Scotia, tidal bores of more than a meter in height are found. Another example of a solitary wave in water is that which occurs between two layers of water at different temperatures. The waves, known as "internal" solitary waves, propagate along the boundary, with the warmer upper layer penetrating up to ten meters below the normal boundary between the two layers and move slowly while carrying a significant amount of energy. These waves cause surface striations that have been photographed, especially in the Andaman Sea off the coast of Thailand. Tsunami, or tidal waves, are probably solitary waves as well. These waves propagate for hundreds or thousands of miles across the open ocean with relatively small amplitude, but as they approach land, the nonlinearity caused by interaction of the wave with the ocean floor begins to increase, causing the wave to steepen and narrow into a destructive wall of water that strikes land.

A second important application of solitary waves is to the propagation of light pulses in optical fibers. To be useful for transmitting information, the waves used must be complex. Sinusoidal waves can carry only two pieces of information, the frequency and amplitude. Complex waves are more useful, since they can be modulated and in so doing can carry more information. Such complex waves are subject to dispersion, however, in addition to the inevitable attenuation of the signal caused by absorption and light loss from the sides of the fiber. Even if the loss of amplitude from dispersion did not occur, the mere spreading of the wave pulses would limit the information-transmitting capacity of the fiber, since the adjacent pulses begin to overlap as a result of dispersion. The use of optical solitary wave pulses to circumvent these difficulties was proposed in 1973 by Akira Hasegawa and Fred Tappert of Bell Labs; at that time, however, the intensity of the light pulse needed to achieve the required nonlinear interaction between the wave pulses and the optical-fiber medium was not available in the frequency region where other fiber losses were minimal. In 1980, Linn Mollenauer of Bell Labs developed the necessary infrared soliton laser and, with colleague Roger Stollen, successfully demonstrated the transmission of solitary waves through optical fibers. Solitary-wave laser pulses have transmitted data over a single optical fiber at the rate of 10 billion pulses per second (10 gigabits) over distances of more than 20,000 kilometers without error. This is equivalent to transmitting several thousand simultaneous telephone conversations over a single optical fiber.

Another important area in which solitary waves are involved is in the field of solid-state physics. In fact, current interest in solitary waves was renewed as a result of investigations into solid-state phenomena undertaken by Enrico Fermi, John Pasta, and Stanislaw Ulam in 1955. They were using one of the first generation of electronic computers to study the conduction of heat through solids. The solid was modeled as a set of masses, connected by springs to represent the forces between molecules. The springs were treated as slightly nonlinear to account for the observed finite thermal conductivity of solids. The periodicity of the network of masses meant that the model system would be dispersive, with the amount of dispersion depending on the difference between the wavelength of a particular wave and the distance between masses. Having started the system out with the thermal energy concentrated in a few modes of the system, they expected to see the energy become gradually dispersed throughout all the modes of the system. Instead, the simulation showed that the energy flowed back and forth through all the modes of the system and then reconcentrated near the initial state, much as in the case of a solitary wave where the dispersion and nonlinearity adjust to a balanced situation. In 1976, Norman Zabusky and Martin Kruskal refined the approximation of the solid, treating it as a continuous medium by letting the spacing between the masses approach zero. The equation that resulted for the propagation of waves through the solid was the Korteweg-deVries equation, which had earlier been shown to describe the propagation of solitary waves in water. Subsequently, Zabusky and Kruskal discovered a number of complex phenomena in solids, but among the most unexpected was the behavior of two solitary waves upon collision. They found that under certain circumstances the two solitary waves emerged unchanged from the collision, except for a phase shift. This particle-like behavior was the basis for naming these special solitary waves "solitons." Modern analysis and understanding of solids makes extensive use of the soliton concept, and with the development of high-intensity lasers, it is possible to introduce solitary heat waves directly into solids.

Context

The first recorded observation of a solitary wave occurred in August, 1843, when a Scottish engineer, John Scott Russell, witnessed a boat being drawn by two horses down the Edinburgh-Glasgow canal. When the boat suddenly stopped, the single wave of water that it produced continued forward without apparent diminution of its size or speed, and with no apparent change in its shape. Russell reported his observation of this "great wave of translation" to the British Association for the Advancement of Science, along with the results of studies he carried out that established a relationship between the speed of the wave, its amplitude, and the depth of the water channel.

Russell's findings were regarded as controversial, since the prevailing opinion at the time was that permanent waves could not exist. Such localized waves, it was thought, would eventually have to dissipate and disappear. However, in 1849, the Irish physicist George Stokes, who had been one of Russell's initial critics, showed that a solitary wave could arise from a combination of periodic waves containing nonlinear terms. Yet the first step in truly understanding solitary waves did not come until the 1870's, when the French theoretical physicist Joseph Boussinesq and the English physicist John Rayleigh independently constructed mathematical descriptions of the solitary wave profile and speed from the equations of fluid dynamics.

The final step in the understanding of the particular type of solitary wave witnessed by Russell was made by the physicist Diederik Korteweg and his student Gustav deVries, both Dutchmen, in 1895. Korteweg and deVries found a wave equation satisfied by solitary water waves, in the case where the depth of the water is small compared to the width of the wave. The Korteweg-deVries (KdV) equation relates the amplitude of the wave and changes in the amplitude with position to the change of the amplitude with time. Indeed, the KdV equation contains one term that involves the product of the amplitude of the wave and its rate of change with position, which introduces a nonlinearity. There is also a second term that depends on the rate at which the curvature of the water surface changes with position. Since this quantity will be different for different-frequency waves, this term is responsible for introducing dispersion. Thus, the KdV equation does include mathematical descriptions of the two prerequisites for solitary waves, dispersion and nonlinearity.

To make practical use of the KdV equation requires that solutions of the equation be found. One such solution had already been found earlier by Boussinesq and Rayleigh. That particular solution describes a solitary wave that moves with a speed equal to twice its amplitude. At this time, however, there existed no general methods for searching for solutions of equations such as the KdV equation, and any approximate solutions obtained by numerical methods would have to wait for the development of computers. For these reasons, the study of solitary waves became relatively dormant until it was revived by the findings of Fermi, Ulam, and Pasta in the 1950's.

Principal terms

DISPERSION: The change in the shape of a waveform caused by the varying speeds of the different frequency waves that make up the waveform

INTERFERENCE: The result of combining two or more waves; in constructive interference, the result is larger than either wave separately; in destructive interference, the result is smaller than either wave separately

LINEAR: A situation in which the response of a system is linearly proportional to the applied displacing force

NONLINEAR: A situation in which the response of a system is not linearly proportional to the applied displacing force

OPTICAL FIBER: An optically transparent fiber through which a light wave is propagated with very little loss of light energy through the sides of the fiber

PHASE SHIFT: A relative shift in the position of a given point on two waves, for example, the distance between the peaks of two waves

WAVE EQUATION: An equation that describes the behavior of a wave in space and time; the equation of a particular wave is determined by the type of wave and its interaction with the medium through which the wave propagates

WAVEFORM: The shape of the wave profile at any instant of time

Bibliography

Herman, Russell. "Solitary Waves." American Scientist 80 (July/August, 1992): 350-361. An excellent review article that describes the distinctions between ordinary waves and solitary waves, as well as the history of the development of the understanding of solitary waves. Also has a bibliography of more than a dozen useful sources.

Infeld, Eryk, and George Rowlands. Nonlinear Waves, Solitons, and Chaos. New York: Cambridge University Press, 1990. This unusual book contains much advanced material, but it is liberally interspersed with excellent basic descriptions of solitary waves and solitons as they appear in nature. The diligent reader can find much useful information here. Also contains an excellent list of original sources.

Main, Iain G. Vibrations and Waves in Physics. 2d ed. New York: Cambridge University Press, 1984. Contains very good discussion of most aspects of wave motion in general, including dispersion and nonlinearity. Chapter 16 is specifically devoted to solitary waves.

National Research Council. Physics Through the 1990's: Condensed-Matter Physics. Washington, D.C.: National Academy Press, 1986. One in a series of books that survey the state of the field for various areas in physics. The various roles of solitons in condensed-matter physics are briefly described, with particular emphasis on possible future applications of solitons in this area. Specifically, the use of solitons in understanding phase transitions and properties of defects in solids are considered.

‗‗‗‗. Physics Through the 1990's: Plasmas and Fluids. Washington, D.C.: National Academy Press, 1986. The various roles of solitons in plasma physics and fluid physics are briefly described, with particular emphasis on possible future applications of solitons in these areas. Focuses on the use of solitons in understanding nonlinear phenomena in plasmas.

Rebbi, Claudio. "Solitons." Scientific American 240 (February, 1979): 92-116. Focuses on the application of the soliton concept to the modeling of elementary particles but does contain a nice summary of the solitary wave. Includes a number of excellent diagrams that illustrate the collision of solitons and a good survey of the development of the concept of a topological soliton.

Saleh, B. E. A., and M. C. Teich. Fundamentals of Photonics. New York: John Wiley & Sons, 1991. A comprehensive text dealing with pure and applied optics. Although it is of a somewhat technical nature, it contains excellent diagrams and concise descriptions. The quantitative level of the text is appropriate for the advanced undergraduate, but the features mentioned above should make it useful, with some effort, to other readers. In particular, the treatment of nonlinear optics principles is useful, and the descriptions of spatial solitons and self-focused optical beams are very good.