Lone Wave

At the heart of new research into proteins, dark matter, and particle physics is a peculiar sort of ripple that defies all expectations -- a wave that doesn't wave.

By David H. Freedman|Thursday, December 01, 1994
For a mathematician, Jerry Bona spends a lot of time perfecting a purely physical skill. To demonstrate his prowess, he grabs a plastic plate and purposefully swishes it through the four inches of water that sits in the singular glass canal dominating his laboratory. The canal has the height and depth of a large home aquarium, but it runs the entire length of the 60-foot wall to which it is bolted with bright yellow metal ribs. Bona's swish produces a single swell of water, about two feet long and a few inches high. It sails down the enclosure 60 feet to the other end, crashes into the glass wall, then bounces back to begin the return trip, looking none the worse for wear.

"Not bad, huh?" says Bona, gazing with obvious affection at the wave, which continues to rebound through the odd water tank, back and forth, again and again, keeping its shape and most of its size. It takes several minutes before the water is fairly still again. "Most mathematicians wouldn't walk two blocks to see an experiment, much less build their own laboratory," he jokes.

The $100,000 water tank is the pride and joy of Bona's Penn State lab, and he enjoys showing it off. It was constructed two years ago in what used to be a women's dormitory, a modest building sheltered by the last remaining large stand of elm trees in North America. The tank is small by public-aquarium standards--though a bored student once sneaked goldfish into it--but what it lacks in volume it makes up for in precision: adjustable supports straighten the walls of the tank to a tolerance of a ten-thousandth of an inch to avoid energy-robbing and shape-deforming bends, and the water is distilled, filtered, and skimmed to remove dust, which can also interfere with the waves. The computer-controlled wave generator is still being perfected, so for now Bona makes do with his plate, having more or less mastered the fine art of producing waves manually. "It takes some practice," he admits.

It's not just any wave that Bona has sweated to perfect. The wave that bounces off walls with impunity and refuses to fade is known as a soliton. Bona studies the behavior of solitons so that he can come up with better mathematical tools for analyzing these strange waves--tools of great interest to physicists, biologists, and other scientists. Why would scientists want to analyze solitons? Because the immortal wave has turned up almost everywhere researchers have looked for it, from the behavior of complex biological molecules to gigantic, bizarre conglomerations of matter that might be lurking in or between galaxies. "Solitons haven't turned up in theology or sociology yet," says Alan Newell, a University of Arizona physicist specializing in the durable lone wave, "but they're just about everywhere else."

The ubiquitous solitons seem to be so important, in fact, that an obvious question arises: Why didn't anyone notice them a long time ago?

Actually, somebody did. Nearly 160 years ago a heavily loaded barge being pulled by horses down the Union Canal near Edinburgh, Scotland, was suddenly halted by a broken rope. As the barge settled into the water, it released a large, single lump of a wave that headed down the canal at about nine miles per hour and, as it traveled, showed no signs of weakening. Watching openmouthed at the canal's edge was John Scott Russell, a young engineer who had been hired by the barge company to determine the practicality of switching from horsepower to steam power. Russell knew enough about water waves to realize that there wasn't supposed to be any such thing as a lone, massive wave that kept its size and shape. Waves were supposed to break up, flatten out, crash; they were supposed to, well, wave. So he jumped on his horse and followed the wave for nearly two miles before losing it around a bend.

Russell's encounter left him obsessed with the lone wave. He soon figured out how to generate similar waves purposely with the canal barge, and he tried to convince the world that these waves were unlike any others ever studied. Scientists scoffed. English Royal Astronomer George Biddel Airy loudly proclaimed that the lone wave was nothing more than the top half of an ordinary wave. The celebrated Lord Stokes produced a mathematical proof that Russell's wave was a physical impossibility.

It's easy to see why solitons were considered unlikely. A wave occurs when a medium is somehow disturbed at a particular location--a stiff wind causes water to pile up into a hump; a vibrating drumhead pushes together the air molecules directly above it. That disturbance then moves off through the medium as the displaced particles pull or push on their neighbors and cause them to be displaced in turn. Thus the wind-driven hump in the water heads off as a water wave, and the jam-up in the air molecules speeds along as a sound wave.

The speed at which waves travel can be affected by their shape and wavelength, or the distance between one crest of the wave and the next. In most familiar media, such as air or water, long-wavelength (low- frequency) waves travel faster than short-wavelength (high-frequency) waves. This makes intuitive sense: a long wave has to "wave" fewer times to cover a given distance than does a short wave.

In nature, waves are rarely generated by a single, simple, isolated disturbance. As a consequence, waves tend to include a mixture of short and long wavelengths. Because the longer wavelengths tend to race ahead of the shorter ones, the wave spreads out in much the way that a tight pack of runners spreads out as faster runners pull in front of the slower runners behind them. This spreading effect, which eventually causes a wave to peter out to nothingness, is called dispersion. It is dispersion that causes the small wave generated by splashing your hand on the surface of a swimming pool to fade out before it gets very far.

Often another effect takes its toll on the integrity of the wave, distorting it not by dispersion but by compression: instead of the different wavelengths separating themselves out by speed, the tallest part of a single hump moves up front and overtakes the leading edge. As a result, the hump steepens and narrows. This compressing effect is especially prevalent in situations such as that presented by the sloping shoreline of an ocean. The shallow bottom provides a frictionlike resistance that causes the wave to narrow, much as a loose pack of runners bunches up if the front-runners suddenly encounter rough terrain. That's why an ocean wave approaching the shore rises and steepens so much that it often breaks.

For centuries physicists rejected the idea of a persistent, unchanging wave because it seemed that between the distorting effects of dispersion and compression, a wave had little opportunity to keep its shape. It had never occurred to anyone that the two effects might, under the right conditions, exactly balance out. That is, the tendency of the wave to spread might be exactly canceled by the tendency of the wave to narrow. The result would be the Energizer bunny of the wave world: it would just keep on going and going and going. That's exactly what happened with Russell's canal wave. It maintained itself because the sides and bottom of the canal provided just enough of a compressing force to counter the normal dispersion of a wave.

Airy and Stokes--and Russell himself, for that matter--had all failed to grasp that enduring waves could emerge from this balance. In 1895 the Dutch mathematicians Diederik Johannes Korteweg and Hendrik de Vries recognized that such a balance of dispersion and compression could in fact produce a lone, durable lump of a wave, but they believed it could stem only from a highly unusual set of circumstances that would rarely occur in the real world. It wasn't until 1965 that Martin Kruskal of Princeton and Norman Zabusky of Bell Laboratories realized that rather than being freaks of nature, these "solitons," as the two mathematicians dubbed them, appeared to be the rule. In fact, there seemed to be no way to avoid them, as scientists discovered more and more kinds of wave-carrying media--from air to water to electromagnetic fields--that under certain conditions had the right balance of dispersing and compressing effects. Solitons, they predicted, would keep turning up in nature, perhaps where they were least expected.

One place no one had ever thought to look for solitons was in proteins, the complex molecules of carbon, hydrogen, nitrogen, and oxygen that perform virtually all the key functions of cells. Proteins grab molecules and assemble them into cellular structures, tear molecules apart to get at energy, and transport oxygen and other needed elements from one cell to another. They carry out these functions much as complex marionettes would: jerking, stretching, flipping, and twisting into whatever shapes are required for the job. But the details of these transformations are almost a complete mystery to molecular biologists. "Biologists' understanding of how proteins function is a lot like your and my understanding of how a car works," says Alwyn Scott, a University of Arizona mathematician with an intense interest in molecular biology. "We know you put in gas, and the gas is burned to make things turn, but the details are all pretty vague."

Take muscle movement, one of the most basic and important functions in all animals. Muscle is composed of thick bundles of a fibrous protein called myosin that are interwoven, much like interlocking fingers, with thinner bundles of the protein actin. To make a muscle contract, the myosin bundles somehow pull the actin bundles toward them. Biologists' picture of this action has always gone something like this: the heads of the myosin bundles stretch outward, somehow grab the actin bundles, lean back, release the actin, and then stretch outward again to get a new grip farther down on the actin bundles, repeating this motion over and over until the actin bundles are pulled far forward. "It's been called the 'rowboat' theory, and that's not meant in a complimentary way," says Scott. The problem, he notes, is that the theory doesn't explain what enables the myosin to lean and grab so effectively.

Russian physicist Alexandr Davydov first suggested in the early 1970s that solitons could provide a better explanation. Imagine that a nearby molecule releases a small packet of energy, in some form not yet understood. The jolt might cause a momentary distortion at that end of the protein, perhaps taking the form of a lump, like a swell of water in a canal. According to conventional molecular biology, such a distortion would break up and scatter through the protein in a trillionth of a second because of dispersion, just as throwing a rock into a puddle would create a mishmash of rapidly fading turbulence rather than a single, neat, persistent hump. But Davydov noted that myosin, like many proteins, has long sections consisting primarily of a chain of pairs of carbon and oxygen atoms--like long rows of dumbbells--and a wave traveling along a chain of such pairs, he showed, would experience a compressing effect.

Davydov concluded that thanks to the tendency of this effect to narrow any wave passing through the chain, in opposition to the protein's tendency to disperse any waves, a myosin bundle was a perfect medium for a soliton. An initial lump created at one end of a bundle would propagate down, pig-in-a-python-style, without shrinking or losing its shape. When the lump encountered the actin fibers surrounding the myosin bundles, it would push them down, just as a pig-size lump in a python might bend the blades of grass alongside the stationary snake as the lump made its way toward the tail. "It's hard to imagine that some process like this isn't taking place," says Scott. "It's a very efficient way to concentrate energy and get it to the right place."

Scott believes solitons solve other protein mysteries as well. Brain cells, for example, can pass on signals to neighboring brain cells when proteins snag sodium atoms floating inside the cell and pump them outside the cell. But no one had a workable model for this pumping action until Scott and other researchers proposed that the charged particles could be "surfing" on solitons moving through the protein--again, pig-in-a- python-style except with the pig moving on top of the python. Related research is exploring a possible role for solitons in DNA, which contains the blueprints for constructing proteins. Since DNA normally exists as a tightly bound double helix, before other molecules can read the blueprint, the two molecular strands have to be temporarily pried apart, or "unzipped." The mechanism for this splitting has long been a puzzle, but Scott notes that a soliton could travel down the DNA, swelling the molecule to the point of unzipping as it passes by, and then letting the DNA rezip itself after it has moved on. "It would be like a pig in a python, where the python's skin comes apart," he says.

If molecular biologists were surprised to hear that submicroscopic solitons might explain some of the most basic mysteries in their field, imagine how startled astrophysicists and cosmologists were to find that solitons might answer two of the questions that have long haunted them: Are giant black holes the only possible source of the phenomenally huge levels of energy streaming out of the center of some galaxies? And what is the nature of the mysterious "dark matter" that seems to be tugging on galaxies throughout the universe?

The link to solitons was suggested in the late 1970s, when Columbia physicists Richard Friedberg and Tsung-Dao Lee wondered if the waves could play a role in particle physics. They had an idea for the source of the compressive force: the vacuum of space. To a physicist, empty space is never really empty; it always contains a certain minimum amount of energy, which takes the form of a teeming sea of "virtual" particles-- particles that spring into existence out of nothingness and then instantaneously disappear. Calculations suggest that pockets of space could at times arise with even more than this minimal amount of energy. But even if such a higher-energy pocket of empty space were somehow to spring up, it would instantly shrink like a leaking balloon as its extra energy dribbled out into the lower-energy space around it, until it finally popped out of existence.

However, Friedberg and Lee asked themselves what might happen if somehow real particles, instead of just virtual ones, appeared inside such a shrinking bubble of higher-energy space--suppose, for example, you had a few real quarks, the particles that make up protons and neutrons. The bubble would be shrinking, and the quarks trapped inside it would be squeezed closer and closer together. That would provide the compressing effect needed for a soliton. When the quarks were squeezed as close together as they could get, they'd start repelling one another. At this point the compressive effect of the bubble's shrinking would be exactly balanced by the spreading, dispersive push of the confined quarks. The result would be a soliton consisting of unbound quarks trapped inside the bubble.

As Friedberg and Lee continued to explore their theoretical construct, they came across another surprise: there didn't seem to be any obvious limit on how many quarks could be trapped inside a high-energy space bubble, as long as the bubble was big enough. The two physicists, working with Yang Pang of Brookhaven National Laboratory, decided to push their strange new solitons to the limit. "We wanted to see how big we could make these things," says Friedberg with a grin. The answer, when it was published, shocked physicists. Friedberg, Lee, and Pang found that their odd soliton could be as large as several light-years across, or the size and mass of a million billion suns. "We never got them quite as big as a galaxy," Friedberg says with a shrug.

Strangest of all was that the three scientists had constructed this bizarre theoretical entity without relying on the one phenomenon that holds together all other celestial objects: gravity. In fact, according to their calculations, the quarks in the space bubble wouldn't exert enough of a pull on one another to hold together. But their supermassive "soliton star" didn't need gravity to keep its shape; the quarks could be held together solely by the compression of the higher-energy space bubble.

It's an interesting model, but is there any reason to believe such an object could have arisen in our universe? Physicist Hong-Yee Chiu at the Goddard Space Flight Center came up with a scheme for the star's creation. It reaches back billions of years to shortly after the Big Bang, when energy was coursing through the hot young universe and particles hadn't yet formed. As the universe slowly cooled, the energy levels in space dropped, but not smoothly: like steam condensing into water, the lower-energy space appeared as droplets. Eventually, as the condensation continued, the lower-energy space came to all but fill the universe, and the remaining higher-energy space was left in pockets much like steam bubbles in water.

Around this time in the universe's evolution, quarks were forming. If a high concentration of quarks happened to pop up in a collapsing pocket of high-energy space, Chiu hypothesized, the quarks could have been trapped and squeezed by the pocket and would have served to prop it up--creating a soliton star. Once formed, the star's total mass would be so great that it could feed off surrounding matter, pulling it in and then swallowing the matter's protons and neutrons. Inside the higher-energy space of the star, the protons and neutrons would be instantly converted to unbound quarks, and the star would grow.

Though many astrophysicists believe the sources of the energy streaming out of the center of some galaxies--energy equal to that of millions of suns--must be giant black holes, Chiu and others think gigantic soliton stars provide an alternative explanation. The energy would pour out, as protons and neutrons swallowed by the soliton star were converted into free quarks--a kind of subatomic fission. "According to most theories," says Chiu, "black holes can't be bigger than ten times the mass of the sun, because they would expel any extra mass as they were collapsing. But whatever is at the center of these galaxies seems to be on the order of 100 million times the mass of the sun. A soliton star seems to fit the picture."

Different soliton stars might have different properties, and Friedberg did not try to pin down exactly what complete set of properties a given star might have. Inside the star, the laws of nature could be radically different from what we're accustomed to: electrons might each weigh a pound, and light might travel at a tenth its normal speed.

But since a soliton star can in principle be fantastically massive and yet utterly invisible, issuing neither heat nor light, it is a perfect--if exceptionally exotic--candidate for dark matter, the stuff that physicists' calculations tell them must be providing perhaps 90 percent of the universe's mass but which appears to be undetectable by normal means. Though most physicists searching for dark matter think their quarry is likely to turn out to be some sort of small particle, Friedberg notes that soliton stars could fill the bill. "Dark matter is up for grabs right now," he says. "Who knows?"

As provocative and potentially significant as solitons have proved to the study of proteins and the missing matter in the universe, these applications may turn out to be simply part of a flood of crucial insights. Solitons have turned up in theories of how earthquakes release destructive energy and how weather patterns form. Planetary astronomers have proposed that Jupiter's red spot is a soliton amid the slow-moving turbulence of the planet's atmosphere. Solitons of magnetism have been observed and may someday turn up on computer chips as a useful way to store a piece of information. Laser scientists have developed devices that produce a soliton whose energy is released at a distance from the laser; such a wave could be used to vaporize diseased tissue in the body without even having to break the skin above it. Boojums, compactons, fluxons, antikinks, twists--all are names for the types of solitons predicted in everything from supercold fluids to empty space.

Solitons even offer a simple, intuitive means for smoothing out the otherwise schizophrenic nature of quantum mechanical behavior, where matter sometimes acts like a wave and at other times acts like a particle; solitons, of course, offer features of both in one package. No one has yet produced a field theory in which all particles can be seen as solitons, but neither are physicists ready to rule such a theory out. "Solitons look a lot like what people have always thought a particle should look like in quantum mechanics," says Alan Newell.

Even if solitons don't give physics a Theory of Everything, their ubiquity and influence suggest that there are hidden connections between different realms of nature. "This shows that no area of science stands alone," says Newell. "The soliton cuts across all of them."
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