# New (and Old) Wave Math

Martin Kruskal (1925-2006) worked on a wide range of problems in pure and applied mathematics, but is best known for the discovery of solitons. Made over the course of a decade in collaboration with Norman Zabusky, Robert Miura, John Greene, and Clifford Gardner, the discovery began with an analysis of the now-famous Fermi-Pasta-Ulam problem. In a numerical experiment, the researchers sought to gauge the effect of weak nonlinearity in a system of coupled harmonic oscillators. They had expected the system to “thermalize,” with energy in the lowest Fourier mode flowing irreversibly into higher modes—indeed, the original idea for the experiment was to study the rate of thermalization. Instead, they found that the energy never went beyond the first few modes, and reverted to the first mode in almost periodic fashion.

Kruskal and Zabusky found that the continuum limit of the FPU system led to the Korteweg–de Vries equation, already familiar from the theory of uni-directional shallow water waves. The KdV equation had originally been introduced to account for the existence of “solitary” waves, famously first observed in 1834 by the Scottish engineer John Scott Russell. It has a simple solitary wave solution in the form of a hyperbolic secant whose speed varies with its amplitude: The taller the wave, the faster it goes. But Kruskal and Zabusky found additional gold in the KdV equation.

In a four-page paper published in *Physical Review Letters* in 1965, they reported the results of their own numerical experiments, in which they found that a single-crested cosine wave (in a domain with periodic boundary conditions) quickly decomposed into a train of solitary waves of different heights and, hence, different speeds. (Technically speaking, the hyperbolic secant is not a solution of the KdV equation on a finite interval with periodic boundary conditions, but only the greatest of sticklers—like Kruskal—ever sweats the exponentially small stuff.) Because of their varying heights, the solitary waves—there were seven of them in the numerical experiment—traveled at different speeds, and thus had to interact as they traveled around and around the periodic domain. That’s where the real surprise popped up and solitons earned their name: Instead of merging like raindrops or shattering like glass balls, the solitary waves emerged from the collisions intact.

It’s easy to be blasé these days about nonlinear waves that interact like particles, but it was an eye-opener in the 1960s. In their write-up, Kruskal and Zabusky underlined the key observation: “Here we have a nonlinear physical process in which interacting localized pulses do not scatter irreversibly.”

Kruskal realized that conserved quantities had to be lurking within the KdV equation. In fact, there are infinitely many. The KdV equation, moreover, turned out to be prototypical in this regard: Kruskal and colleagues started finding integrable systems under virtually every nonlinear rock they examined. The theory grew from a cottage industry to one of the main themes of modern mathematical physics.

*To learn more about Kruskal and particle physics, see "A Submervise Model of Particle Physics," published in the same issue.*