English

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Introduction to Partial Differential Equations

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Solitons are waves with exceptional stability properties which appear in many areas of physics. The basic principles of solitons are introduced here. A soliton is a solution of a nonlinear partial differential equation which represent a solitary traveling wave.

History – Key persons/Experiments
- John Scott Russel describes his wave of translation. The discovery in described here in Russell’s own words
‘’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".

Motivation – impact of the method on society and overview of applications

Today solitons take apart in many physicals areas, like hydrodynamics, quantum mechanics, particle physics and so on.
It was found a lots of equation with soliton-type solutions.
An example: High-speed optical communications require intense, short, spectrally pure light pulses, such a solitary solution of a non-linear form of the Schrödinger equation.

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Solitons are waves with exceptional stability properties which appear in many areas of physics. The basic principles of solitons are introduced here

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Physics of Solitons, Thierry Dauxois
Solitons : An introduction, P. G. Drazin, R. S. Johnson, E. J. Hinch

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