# Focus: *Landmarks*—Computer Simulations Led to Discovery of Solitons

*Landmarks**articles feature important papers from the archives of the* Physical Review *journals.*

The emergence of computers as a tool for doing science didn’t merely help researchers solve difficult problems. It also led to the discovery of entirely unexpected phenomena. An example appeared in *Physical Review Letters* in 1965, in a report describing solitary waves—dubbed “solitons” by the authors—that moved somewhat like individual particles. Solitons are now recognized as a widespread phenomenon, occurring in fluids, optics, and even the ultracold atomic clouds known as Bose-Einstein condensates.

In 1955 Enrico Fermi, John Pasta, and Stanislaw Ulam (FPU) came across a puzzling result when using the MANIAC I computer at what was then called the Los Alamos Scientific Laboratory in New Mexico. They wrote a program to follow the motion of up to 64 masses connected by springs in a horizontal line. Each mass could move only in the direction of the line, stretching or compressing the two springs connected to it.

The team started the simulation by displacing each mass from its initial position in a pattern that formed one half of a sine wave, with the end masses having zero displacement and the middle masses having the greatest displacement. The masses would then oscillate, and if the springs were strictly linear—that is, if their force were proportional to the amount of stretch or compression—then a snapshot of the motion at any time in the future would show the masses still in a sine wave pattern. But Fermi and his colleagues added a small degree of nonlinearity to the springs’ force, expecting it to break up the sine wave and cause the oscillation energy to become, in time, equally distributed among all the masses.

That’s not what happened. Although the sine wave indeed evolved into a more complex oscillation, the motion of the masses never became completely disorderly, and in fact it periodically returned to the initial state [1].

A decade later, Norman Zabusky, then at Bell Labs in Whippany, New Jersey, collaborated with Martin Kruskal of Princeton University to re-examine the FPU work. They transformed the discrete masses-and-springs equation into one for a continuous system similar to water waves. The team then programmed a computer to calculate the wave motion over a fixed horizontal distance but in such a way that a disturbance passing out of one side of the range reappeared at the other side.

Like FPU, Zabusky and Kruskal set the system in motion with an initial sine wave pattern. As the wave rolled along, its leading edge became steadily steeper and then developed smaller-wavelength ripples. These ripples eventually grew into individual waves that moved independently, with a velocity that depended on their height. Remarkably, when these new waves occasionally collided, they passed through each other, emerging almost wholly unscathed from their encounters. In addition, the waves would regularly align to reproduce the initial sine wave, momentarily, before separating again and repeating—not quite perfectly—the same cycle. This phenomenon was similar to the periodic return to the initial state that FPU had observed.

Zabusky and Kruskal soon learned that their equation had a name: It was the Korteweg-de Vries equation, devised in 1895 by two Dutch physicists to explain solitary waves—single, isolated peaks—that had been observed occasionally in canals as early as the 1830s. Any interaction between such waves was expected to be complex and hard to predict because of the nonlinearity. However, Zabusky, now at the Weizmann Institute of Science in Rehovoth, Israel, says that he and Kruskal were amazed to find that their solitary waves could run through each other and remain intact. He says this behavior was so striking that these waves deserved a name, and he came up with “solitons.”

Zabusky says that the discovery of solitons met with some skepticism, which he was able to overcome in part by using Bell Labs’ facilities to make movies of the waves. Some years later, physicists began to find soliton solutions in other wave equations, making their credibility unarguable.

Gennady El of Loughborough University in England calls the Zabusky-Kruskal paper a classic example of how numerical simulations can “provide insight into deep and fundamental properties of a mathematical model and lead to the discovery of completely new phenomena.” An important modern offshoot, he adds, is the theory of “dispersive shock waves”—coherent, nonlinear structures that can be regarded as systems of interacting solitons, and which show up in Bose-Einstein condensates and nonlinear optics.

–David Lindley

David Lindley is a freelance writer in Alexandria, Virginia, and author of *Uncertainty: Einstein, Heisenberg, Bohr, and the Struggle for the Soul of Science* (Doubleday, 2007).

## References

- E. Fermi, J. R. Pasta, and S. Ulam, “Studies of nonlinear problems. I.,” Report LA-1940, Los Alamos Scientific Laboratory (1955)