✎ Solitons [The Physics Travel Guide]

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====== Solitons ======

[[advanced_tools:feynman_diagrams|Feynman diagrams]] do not describe everything that can happen in a [[theories:quantum_theory:quantum_field_theory|quantum field theory]]. There can be classical solutions of the field equations that describe larger lumps of field excitations that aren't describable by Feynman diagrams.

Classical solutions of the field equations with finite energy are called solitons.

Such solutions are important to describe, for example, the vacuum of a theory. A famous example is the [[advanced_notions:quantum_field_theory:cd_vacuum|QCD vacuum]] which can only understand with the help of [[advanced_notions:quantum_field_theory:instantons|instantons]].

In addition, there is an [[http://www.pbs.org/wgbh/nova/blogs/physics/2011/12/beautiful-losers-kelvins-vortex-atoms/|old dream]] that all elementary particles could be explained as topological solitons. (There are lots of problems with this idea, but at least, [[https://plus.google.com/+UrsSchreiber/posts/Z2LfHsyxgR8|instantons come somewhat close]].)

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Important examples of solitons are:

* [[advanced_notions:topological_defects:magnetic_monopoles|Magnetic Monopoles]] - usually in 3+1 dimensions
  * [[advanced_notions:quantum_field_theory:sphalerons|Sphalerons]] - usually in 3+1 dimensions
  * [[advanced_notions:quantum_field_theory:instantons|Instantons]] - usually in 4 dimensions
  * [[advanced_notions:topological_defects:vortices|Vortices]] - usually in 2+1 dimensions
  * [[advanced_notions:quantum_field_theory:kinks|Kinks]] - usually in 1+1 dimensions

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Soliton equations are special, but they are by no means unimportant; they arise in such diverse areas of applied science as hydrodynamics, nonlinear optics, plasma dynamics, meteorology, Josephson type superconductive devices, nonlinear acoustics, oceanography, domain wall dynamics, dislocation theory, nonlinear electric filters, and in the theory of elementary particles.

<cite>http://garfield.library.upenn.edu/classics1979/A1979HF81700001.pdf</cite>
</blockquote>

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<tabbox Layman> 
{{ :advanced_notions:quantum_field_theory:solitonicwavevsnormal.png?nolink&600|}}

A soliton is a curious type of __stable__ wave. Usually, waves vanish after some time. Think of a stone that gets thrown into some water. The stone will cause waves. At first, these waves will be large, but quickly get smaller and smaller until they are no longer visible. The waves flattens out. This process is known as dissipation.

In contrast, a soliton does not dissipate its energy but remains its form over a long time, possibly infinitely.

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<blockquote>
Before suggesting why [[http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1451226|the paper]] has been so often cited, it is appropriate to explain what the term soliton means. As coined by Zabusky and Kruskal,  this term is generic for special solitary wave solutions of certain nonlinear wave equations.

What then is a solitary wave? It is a pulse-like wave that travels with constant speed and shape; the effects of dispersion on the wave shape are just balanced by those of nonlinearity. There is just enough yin for the yang; it is a dynamically self-sufficient object, a ‘thing.’

Solitons are solitary waves that preserve their speeds and shapes after
mutual collision. They play a role in the
construction of complete solutions for
the nonlinear wave equation that
corresponds to the role played by
Fourier components in the construction
of solutions for linear wave equations.

<cite>http://garfield.library.upenn.edu/classics1979/A1979HF81700001.pdf</cite>
</blockquote>

<blockquote>“A solitary wave travels without changing
its shape, size, or, speed” <cite>J. Scott Russell, “Report on waves”</cite></blockquote>
 
<tabbox Student> 
{{ :advanced_notions:quantum_field_theory:dissipationvsnonlinearitycropped.png?nolink&600|}}
Solitons are stable through the interplay of dissipation and non-linearity of the underlying wave equations.

* Dissipation happens because a wave packet is a superposition of waves with different wave-lengths and usually waves with different wave-lengths travel with different velocities. 
  * Non-linearity of the wave equations can result in waves that get __steeper__ over time. A good example are the waves that can be observed at a beach.

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

--> Example1#

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--> Example2:#

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<tabbox FAQ> 
  
<tabbox History> 
Solitons were first described in 1834 by John Scott Russell:

<blockquote>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.[2] <cite></cite></blockquote>
</tabbox>

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