Nonlinear Waves in Integrable and Nonintegrable Systems

Excerpt

Wave phenomena are abundant in nature. Familiar examples include water waves and optical waves. Low-amplitude waves are governed by linear partial differential equations. A main feature of linear wave phenomena is dispersion, i.e., different Fourier modes inside a disturbance travel at different speeds. This often leads to the spreading and decay of a local disturbance. For instance, if we throw a stone into a pond, concentric rings of ripples will spread out. Outer ripples generally have higher wavenumbers (shorter spatial periods) and travel faster. After a short time, all the ripples will disperse and disappear (even in the absence of dissipation). Similarly, when a low-intensity light beam passes through the air or a crystal, the beam broadens over distance. This phenomenon is called diffraction in optics, and it is the counterpart of dispersion in water waves.

In 1834, Scott Russell accidentally observed a type of water wave which did not disperse. Below is his original description:

“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth, and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation” (Russell (1844)).

This phenomenon was recreated on the Union Canal near Edinburgh in July 1995, and a photo is shown in Fig. 1.

Russell's observation puzzled physicists for a long time and caused much controversy, because it could not be explained by linear water wave theory.