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Unformatted text preview: Chapter 10 Solitons Starting in the 19 th century, researchers found that certain nonlinear PDEs admit exact solutions in the form of solitary waves, known today as solitons . There’s a famous story of the Scottish engineer, John Scott Russell, who in 1834 observed a hump-shaped disturbance propagating undiminished down a canal. In 1844, he published this observation 1 , writing, “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 was so taken with this phenomenon that subsequent to his discovery he built a thirty foot wave tank in his garden to reproduce the effect, which was precipitated by an initial sudden displacement of water. Russell found empirically that the velocity obeyed v ≃ radicalbig g ( h + u m ) , where h is the average depth of the water and u m is the maximum vertical displacement of the wave. He also found that a sufficiently large initial displacement would generate two solitons, and, remarkably, that solitons can pass through one another undisturbed. It was not until 1890 that Korteweg and deVries published a theory of shallow water waves and obtained a mathematical description of Russell’s soliton. 1 J. S. Russell, Report on Waves, 14 th Meeting of the British Association for the Advancement of Science, pp. 311-390. 1 2 CHAPTER 10. SOLITONS Nonlinear PDEs which admit soliton solutions typically contain two important classes of terms which feed off each other to produce the effect: DISPERSION − ⇀ ↽ − NONLINEARITY The effect of dispersion is to spread out pulses, while the effect of nonlinearities is, often, to draw in the disturbances. We saw this in the case of front propagation, where dispersion led to spreading and nonlinearity to steepening. In the 1970’s it was realized that several of these nonlinear PDEs yield entire families of exact solutions, and not just isolated solitons. These families contain solutions with arbitrary numbers of solitons of varying speeds and amplitudes, and undergoing mutual collisions. The three most studied systems have beencollisions....
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- Fall '11
- Equations, Partial differential equation, Korteweg–de Vries equation, Solitons, soliton