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Solitons: Introduction and Applications (Springer Series in Nonlinear Dynamics) (Muthusamy Lakshmanan)

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“the Yang-Mills equations are nonlinear, therefore there is little hope of finding a closed-form solution.” Such a statement seems plausible. Linear differential equations with constant coefficients are the only differential equations for which a general solution is given in closed form. As often occurs in life, however, the exceptions to the rule are sometimes more interesting than the rules themselves. Let us digress from quantum physics to the motion of water, where British shipbuilder John Scott Russell noticed a solitary wave in a canal in August 1834. Neither Airy nor Stokes accepted this observation, yet in 1895 Korteweg and de Vries found an equation for a wave travelling in shallow water in one direction: u̇ + 6•u•uₓ + uₓₓₓ = 0. The KdV equation is easily solved by restricting from two independent space-time dimensions (x,t) to a single dimension x−λt — a frame matching the speed λ of a travelling wave.

Mikhail Ilʹich Monastyrskiĭ, Riemann, Topology, and Physics

Making waves with metamaterials

For Jordan Raney, assistant professor in the Department of Mechanical Engineering and Applied Mechanics, cutting-edge science sometimes involves whacking a rubber disc with a hammer.

In a recent experiment, he and Chengyang Mo, a graduate student in Raney's Architected Materials Laboratory, used a 3-D-printer to make this unusual, Frisbee-sized structure. It consists of hundreds of connected rubber squares, each with a ball bearing inside. It's an example of a "mechanical metamaterial," a class of systems that exhibit unusual physical behaviors stemming from their internal geometry rather than from the properties of the materials of which they're made.

Raney and Mo were trying to understand the propagation of a unique type of nonlinear wave called a soliton. To do that, they placed their structure on top of ball bearings and set up a high-speed camera to record the wave's motion at 6,000 frames per second. Then, according to their paper, published in the journal Physical Review Letters, they "excited the sample with an impactor"—or hit it with a $6 mallet from the hardware store.

Their study is the first to show how these waves travel in a soft 2-D system.

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 [14 km/h], preserving its original figure some thirty feet [9 m] long and a foot to a foot and a half [30−45 cm] in height. Its height gradually diminished, and after a chase of one or two miles [2–3 km] 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.

—John Scott Russell, 1845

meeting of the British Association for the Advancement of Science, York, September 1844 (London 1845), pp 311-390, Plates XLVII-LVII).

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