# In the OSTI Collections: Solitons

 Article Acknowledgement:Dr. William N. Watson, PhysicistDOE Office of Scientific and Technical Information

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A wave of any kind—a sound wave, a water wave, a light wave, an electron wave—may have a sinusoidal form with a single, well-defined frequency and intensity.  On the other hand, waves like the red one in the figure below may be analyzable into several component sinusoidal waves of different frequencies and intensities.

Figure 1.  Left:  example of a waveform (in red) whose amplitude above or below its average value at each location along a left-to-right axis equals the sum of the amplitudes of six sinusoidal waves of different wavelengths at the same location, five of which (in blue) are visible in the figure.   Where the different components’ amplitudes are mainly above average or below average, the red waveform’s amplitude will also be well above or below average; where the components’ amplitudes largely cancel, the red waveform’s amplitude will be more nearly average.  The red waveform depends not only on the components’ amplitudes and wavelengths, but on how those components are aligned:  with a different alignment, the sums of the components’ amplitudes at each point would be different.  Right:  the red waveform with all six sinusoidal components spread out, with each component’s maximum amplitude exhibited to its right.  (From Wikimedia Commons[Wikimedia]; made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.)

Light waves in a vacuum always have the same speed, whatever the components’ frequencies or intensities are.  This is an exceptional behavior for waves.  Light waves traveling through matter, or most other waves traveling through any medium, have speeds that depend on both their intensities and their frequencies.  Since the form of the wave depends on how the different components line up, the form would change over time as the different components realigned due to traveling at different speeds, even if the components’ intensities didn’t affect how fast they traveled.  The effect of different components moving at different speeds is illustrated below in the animation of Figure 2, which shows a different initial waveform with different components from the waveform of Figure 1.  If the components of Figure 2 moved left to right at the same speed, the waveform would stay the same shape (top of figure); if the components moved at different speeds, the waveform would change (bottom of figure).  The exact change of shape depends on how the components’ speeds vary with frequency:  here the result is gradual elongation of the waveform.

Figure 2.  Whether a waveform stays the same as it travels depends in part on whether the waveform’s different-frequency components travel at the same speed.  If they do, the components’ relative alignment stays the same and the waveform remains constant (top).  If the components’ speeds differ, the waveform will change (bottom).  (From Wikimedia Commons[WikimediaWikimedia]; released by author into the public domain, and usable for any purpose without any conditions unless such conditions are required by law.)

In media for which the speeds of waves also depend on the waves’ intensities, even sinusoidal components will change their shape, since the extreme high and low portions of those waves will travel slower or faster than the portions whose amplitudes are closer to average.

Thus in many media, waves generally change their shape as they travel, even if the medium they travel in has the same composition throughout.  But waves in some media can propagate indefinitely without distortion, if their shapes are such that the dependencies of wave speed on amplitude and frequency cancel each other out.  News that such waves had been observed in water in 1834 was initially received with skepticism, partly because existing theories of wave motion had accounted for the dependence of wave speed on frequency while the dependence on amplitude had not yet been discovered.[SciTech ConnectWikipedia]  Since then, however, waves whose shape is maintained because their frequency and amplitude speed dependencies balance out have been found in many media.  These “solitary waves” or “solitons” continue to be explored, in order to better comprehend the media they exist in and to put the waves to practical use.

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Observation

One state of matter that can support solitons is plasma, a gas at least partly consisting of charged particles.[WikipediaOSTI]  A thin but very important plasma surrounds the earth.  Since plasma affects the shape of the earth’s magnetosphere[Wikipedia], variations in it (like solitons) affect our wireless communications and, more importantly, the magnetosphere’s ability to shield life on earth from charged particles in cosmic radiation by deflecting them away from the earth’s surface.  The European Space Agency’s Cluster II observatories[Wikipedia] have revealed that the plasma contains kilometer-sized solitons that consist of regions with low electron densities and travel along the earth’s magnetic field lines.  Researchers at the University of New Hampshire[SciTech Connect] and the University of Iowa[SciTech Connect] have collaborated in an effort to determine how these solitons are generated, which included experiments at UCLA with plasma solitons in the university’s Large Plasma Device[Wikipedia], which contains plasma in a cylinder 20 meters long and 1 meter wide.  The information available allowed the researchers to identify a possible soliton-generation mechanism.  Since the laboratory solitons turned out to be similar to the solitons found where auroras[Wikipedia] form (and perhaps elsewhere in the magnetosphere), the researchers determined that the magnetospheric solitons may get their start from a particular type of instability among the many instabilities known to affect plasmas’ equilibrium.  However, the researchers were unable to confirm or disprove the role of this particular instability from Cluster II observatory data because of the limited time resolution of the Cluster II instruments and telemetry.

Kilometer-scale solitons of low electron density in plasma have a quite different nature—and practical significance—from the nanometer-scale solitons that can carry charge in “wires” made up of single molecules.  Research into using single molecules as wires has become extensive with the development of polymers that can conduct electricity, but polymer wires in most devices that use them are made, not of one molecule, but of several.  The late-2014 report “The Organic Chemistry of Conducting Polymers”[SciTech Connect], on the other hand, describes one research group’s multiyear effort to determine how wires made of just one molecule conduct electricity (among other investigations).  First, mobile charges have to be put into the polymer molecule.  If an electron is put in by chemical oxidation, its charge can propagate along the molecule in at least two different ways.  One way involves the electron’s negative charge inducing a distortion of the molecule, with the positive charges in the atoms closest to the electron being attracted to it the most.  This distortion will affect different atoms as the distortion follows the motion of the extra electron along the molecule’s length.  Such electron-plus-distortion entities act like moving particles and are known as polarons[Wikipedia], whether they occur in polymers or in two- or three-dimensional atomic arrays.  Electrons’ quantum-physical wave nature provides another way for its charge to propagate through a molecule, as a soliton.  The researchers’ experiments with 19 different systems showed that polaron conduction was more efficient in polymers whose structure included stable aromatic[Wikipedia] rings of atoms, while soliton conduction was more efficient in aliphatic[Wikipedia] polymers, which lack such rings.

Nanoscale solitons need not be confined to carrying charge across a molecule.  Solitary waves of varying electron-spin orientation could play a role in spintronic devices that exploit the transport of electron spin as well as the transport of electronic charge.[WikipediaOSTI]  Questions about the sort of patterns that might exist in the electrons’ spin orientations were addressed in an investigation reported in the Nature Communications paper “Direct observation and imaging of a spin-wave soliton with p-like symmetry”[DoE PAGES].  Current flowing into a magnetized nickel-iron permalloy sample through a nanometer-sized contact produced spin-wave solitons in the sample, which were observed using x-rays from the Sanford Synchrotron Radiation Lightsource[Wikipedia].  While existing theory had suggested that spin solitons could exhibit the “s-like” pattern of Figure 3c below, in which the electron spins all had a roughly common orientation, it was less clear whether other patterns were possible before the experiments reported in the Nature Communications paper also showed “p-like” patterns, in which electrons on opposite sides of the soliton were oriented in roughly opposite directions.  (The “s”, “p”, and other related designations originated in spectroscopy[Wikipedia] and wound up labeling different wave patterns by way of early atomic physics.)  The investigators inferred that magnetic fields in the electrical contact and the current, combined with the magnetic field applied to the permalloy that gave its electron spins a common orientation axis, produced a region in which a spin wave could localize.  Furthermore, they found that strengthening the applied magnetic field contracted the localization region, thus inducing the soliton electrons’ spin orientations to switch from a “p-like” to an “s-like” pattern and providing a new way to manipulate electron behavior and the information that behavior could encode in a computer or communication device.

Figure 3.  An overview of the experiment described in “Direct observation and imaging of a spin-wave soliton with p-like symmetry”[DoE PAGES]”.  (a)  Schematic diagram.  Circularly polarized x-rays from the Stanford Synchrotron Radiation Lightsource are focused to a spot 35 nanometers wide onto a sample made of nickel-iron (NiFe), copper (Cu), and cobalt-iron (CoFe) layers, in which the copper and cobalt-iron layers are patterned into a 150-nanometer by 50-nanometer ellipse and the nickel-iron (or permalloy) layer is larger. When a magnetic field (H) is applied in the plane of the sample, waves of electron-spin orientation are excited and a direct current (IDC) flows into the nanocontact.  A microwave current (Imw) is superimposed on the direct current to synchronize the spin-wave excitation with the x-ray detection and the Lightsource’s master clock.  Variation of the magnetization is time-resolved along the direction of the x-rays’ propagation.  (b)  X-ray image of the sample’s topography shown with a 200-nanometer scale bar.  (c)  Schematic of two types of spin-wave symmetries.  (After “Direct observation and imaging of a spin-wave soliton with p-like symmetry”[DoE PAGES]”, p. 2.)

Figure 4.  Experimental and simulated results reported in “Direct observation and imaging of a spin-wave soliton with p-like symmetry”[DoE PAGES]”.  (a-f)  Experimental data of angle at which magnetization precessed out of the plane of the sample, as shown by spatial maps 1.5 micrometers wide and 1.5 micrometers high, with successive maps separated by 27 picoseconds.  Magnetic field m0H of 60 milliteslas[Wikipedia] was applied parallel to the horizontal (x) axis.  Nanocontact is indicated by the black ellipse.  A 200-nanometer scale bar is shown in (a).  (g-l)  Simulated maps of the magnetization precession for a 60-millitesla applied magnetic field.  (m-r)  Simulated maps of the magnetization precession for an 80-millitesla applied magnetic field.  Vertical cross sections of the regions indicated by the dashed lines in (ag, and m) highlight differences between “s-like” and “p-like” spin waves.  (After “Direct observation and imaging of a spin-wave soliton with p-like symmetry”[DoE PAGES]”, p. 3.)

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Theory

Whether or not a medium will support solitons is determined by the exact manner in which wave speeds in that medium depend on wave intensities and frequencies.  Many different speed dependencies will work, but the equations that describe such media have one common property:  for the wave speeds to depend on wave intensities, the equation that describes the medium must be nonlinear, meaning that distinct solutions of the equation generally can’t be added together to find another solution.[Wikipedia]  (Sums of linear equations’ solutions can always be added together to get another solution.)  A Los Alamos National Laboratory slide presentation entitled “Solitary waves in nonlinear Dirac equation:  from field theory to Dirac materials”[SciTech Connect] mentions several nonlinear equations and the solitons that some of their solutions describe, but focuses mainly on two nonlinear generalizations of a linear equation[Wikipedia] discovered by Paul Dirac[Wikipedia] in the late 1920s.  While Dirac’s equation described quantum-physical features of electron behavior more accurately other previously-used equations, the nonlinear generalizations model how the collective motion of large numbers of electrons in two different kinds of material resembles the motion of a few lower-mass particles in a vacuum.[arXivOSTI]  Some of the collective motions constitute solitons.  The research presented dealt with conditions under which solitons in the two kinds of material are stable, but the researchers found that three different criteria expected to indicate stability actually implied mutually contradictory conditions, which posed a new problem yet to be solved.

One of the nonlinear equations briefly mentioned in the slide presentation was the primary focus of a set of lecture notes[SciTech Connect] prepared for the United States Particle Accelerator School held in June 2012.  These notes explain how a Korteweg-de Vries equation[Wikipedia] can be derived for a different medium—electrical transmission lines that generate the microwaves that power particle accelerators—and how the microwaves generated appear as solitons, whose mathematical descriptions satisfy the equation.  The lecture notes end by discussing some further points of current microwave generation technology, but they begin (much as the slide presentation on nonlinear Dirac equations did) by briefly recounting the way solitons were observed in 1834 by engineer John Scott Russell[Wikipedia] as he was conducting experiments on narrowboat[Wikipedia] hull designs.  The lecture notes also describe how subsequent experiments and theoretical analysis provided the data and understanding to overcome initial skepticism about solitons’ reality.  Like the slide presentation, the lecture notes also provide illustrations of several known or possible soliton phenomena—in this case Morning Glory clouds[Wikipedia], Jupiter’s Great Red Spot[Wikipedia], paired water vortices[Physics Central], light pulses in optical fibers[Wikipedia], and tidal bores[Wikipedia] in rivers with funnel-shaped estuaries.  The last example is cited as an analog of microwave formation by particle accelerators’ power transmission lines:  while water waves and voltage waves are different things, Kortweg-de Vries equations describe both.  In both cases an initial wave (a high tide or a voltage pulse) travels along a path and breaks up into a series of smaller, closely spaced solitons, thus converting a powerful lower-frequency wave into a train of waves with much higher frequency.

The main equation used before Dirac’s to describe electrons’ quantum-mechanical behavior was the simpler Schrödinger equation—which is still used to analyze situations in which the interrelation of space and time described by relativity theory are not significant enough to warrant dealing with the Dirac equation’s complexity.  Like Dirac’s equation or variants thereof, Schrödinger’s equation or modified versions of it also describe things other than electrons that behave similarly.  A modified, nonlinear Schrödinger equation describes the atoms of certain gases that, when cooled to billionths of degrees above absolute zero, condense into a state in which the lowest-energy atoms share a common behavior rather than behaving differently as they would in a warmer gas.   Certain mathematical functions of position and time that solve the Schrödinger equation represent different possible common behaviors of all the condensate atoms.  Since such a function’s amplitude at any position is the square root of the condensate’s density there, the function represents how density fluctuations move as waves through the condensate (which is named a Bose-Einstein condensate[Wikipedia] after the physicists whose work showed such condensates could exist).  And since the forces on the condensate atoms make the waves’ speed depend in an appropriate way on their amplitude and frequency, the nonlinear Schrödinger equation that represents those forces has solutions that represent solitons.

The nature of the forces at work in a medium determine what kinds of solitons that medium can support.  Among the possibilities for light waves in certain materials are so-called “dark”, “bright”, and “dark-bright” solitons.  “Dark” and “bright” solitons represent mobile volumes of space in which the light intensity was respectively less than and greater than in the space around them.  Analogous “dark” and “bright” solitons can also exist in Bose-Einstein condensates as moving volumes of lower- and higher-than-average atom density.  In gases with two kinds of atoms instead of one, “dark-bright” solitons can exist with lower-than-average density for one kind of atom and higher-than-average density for the other kind—analogously to the optical case of solitary light waves with two different polarizations, one wave “bright” and one “dark”, propagating as a unit.

Reports of mathematical analyses involving all of these soliton types are exemplified by papers like “Stability and tunneling dynamics of a dark-bright soliton pair in a harmonic trap”[DoE PAGES], “Transitions from order to disorder in multiple dark and multiple dark-bright soliton atomic clouds”[DoE PAGES], “Solitons and vortices in two-dimensional discrete nonlinear Schrödinger systems with spatially modulated nonlinearity”[DoE PAGES], and “Effects of interactions on the generalized Hong–Ou–Mandel effect”[DoE PAGES].  The first paper describes a mathematical analysis of the existence, stability properties, and behavior of pairs of dark-bright solitons in a Bose-Einstein condensate whose width and thickness are small and whose length is much longer.  The second discusses how multiple solitons in such a gas can move randomly like gas particles themselves, arrange themselves into an orderly pattern resembling that of atoms in a solid, and transition between the two phases.  The third paper deals with solitons (and vortices) in a Bose-Einstein condensate with relatively large length and width and small thickness.

One use that might be made of these solitons is indicated in the fourth paper.  While that paper’s analysis doesn’t explicitly deal with solitons, it does examine how interacting atoms would exhibit a particular phenomenon that’s potentially useful in quantum computers[OSTIWikipedia] and goes on to note that the stability of bright solitons makes them promising entities for experimenting with the same phenomenon.  The simplest form of this phenomenon, the Hong–Ou–Mandel effect[Wikipedia], is exemplified when two identical particles that are capable of Bose-Einstein condensation enter opposite sides of a device that divides particle beams equally[Wikipedia].  The quantum-physical wave function for the incoming particle beams is indeed equally divided at the beam-splitter’s exit.  However, wave functions for the exiting particles always represent both particles exiting the device on the same side, with a 50:50 chance of exiting at either side given the incoming-particle wavefunction.  The paper deals explicitly with phenomena of the same general type, in which wave functions that represent more than two identical atoms entering an atomic-beam splitter have nonzero probabilities of association with various wave functions for the atoms exiting in certain distributions and zero probabilities for the atoms exiting in others.  If different distributions of exiting atoms were used in a quantum computer to represent distinct sets of bits, the wavefunctions for the dividing input beams would represent all those bit sets simultaneously.  Simultaneous operation on multiple distinct sets of bits as a single entity is a characteristic ability by which a quantum computer could solve certain problems that our “classical” computers, whose individual processors operate on only one set of bits at a time, could never solve quickly enough for the answers to be useful to us.

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References

Wikimedia Commons

Wikipedia

Reports available through OSTI’s SciTech Connect

Reports available through DoE PAGESBeta