- Soliton
In

mathematics andphysics , a**soliton**is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. "Dispersive effects" refer todispersion relation s between the frequency and the speed of the waves. Solitons arise as the solutions of a widespread class of weakly nonlinear dispersivepartial differential equation s describing physical systems. The soliton phenomenon was first described byJohn Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation ".**Definition**A single, consensus definition of a soliton is difficult to find. Drazin and Johnson (1989) ascribe 3 properties to solitons:

# They are of permanent form;

# They are localised within a region;

# They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift.More formal definitions exist, but they require substantial mathematics. Moreover, some scientists use the term "soliton" for phenomena that do not quite have these three properties (for instance, the '

light bullet s' ofnonlinear optics are often called solitons despite losing energy during interaction).**Explanation**Dispersion and non-linearity can interact to produce permanent and localized wave forms. Consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of several different frequencies. Since glass shows dispersion, these different frequencies will travel at different speeds and the shape of the pulse will therefore change over time. However, there is also the non-linear

Kerr effect : therefractive index of a material at a given frequency depends on the light's amplitude or strength. If the pulse has just the right shape, the Kerr effect will exactly cancel the dispersion effect, and the pulse's shape won't change over time: a soliton. Seesoliton (optics) for a more detailed description.Many

exactly solvable model s have soliton solutions, including theKorteweg–de Vries equation , thenonlinear Schrödinger equation , the coupled nonlinear Schrödinger equation, and thesine-Gordon equation . The soliton solutions are typically obtained by means of theinverse scattering transform and owe their stability to theintegrability of the field equations. The mathematical theory of these equations is a broad and very active field of mathematical research.Some types of

tidal bore , a wave phenomenon of a few rivers including theRiver Severn , are 'undular': a wavefront followed by a train of solitons. Other solitons occur as the underseainternal wave s, initiated by seabedtopography , that propagate on the oceanicpycnocline . Atmospheric solitons also exist, such as the Morning Glory Cloud of theGulf of Carpentaria , where pressure solitons travelling in atemperature inversion layer produce vast linearroll cloud s. The recent and not widely acceptedsoliton model inneuroscience proposes to explain the signal conduction withinneuron s as pressure solitons.A

topological soliton , ortopological defect , is any solution of a set ofpartial differential equation s that is stable against decay to the "trivial solution." Soliton stability is due to topological constraints, rather thanintegrability of the field equations. The constraints arise almost always because the differential equations must obey a set ofboundary conditions , and the boundary has a non-trivialhomotopy group , preserved by the differential equations. Thus, the differential equation solutions can be classified intohomotopy class es. There is no continuous transformation that will map a solution in one homotopy class to another. The solutions are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include thescrew dislocation in acrystalline lattice , theDirac string and themagnetic monopole inelectromagnetism , theSkyrmion and theWess-Zumino-Witten model inquantum field theory , andcosmic string s anddomain wall s in cosmology.**History**In 1834,

John Scott Russell describes his "wave of translation ". The discovery is 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". [

*"J. Scott Russell. Report on waves, Fourteenth meeting of the British Association for the Advancement of Science, 1844."*](

**Note:**This passage has been repeated in many papers and books on soliton theory.)(

**Note:**"Translation" here means that there is real mass transport such that water can be transported from one end of the canal to the other end by this "Wave of Translation". Usually there is no real mass transport from one side to another side for ordinary waves.)Russell spent some time making practical and theoretical investigations of these waves, he built wave tanks at his home and noticed some key properties:

* The waves are stable, and can travel over very large distances (normal waves would tend to either flatten out, or steepen and topple over)

* The speed depends on the size of the wave, and its width on the depth of water.

* Unlike normal waves they will never merge — so a small wave is overtaken by a large one, rather than the two combining.

* If a wave is too big for the depth of water, it splits into two, one big and one small.Russell's experimental work seemed at odds with the

Isaac Newton andDaniel Bernoulli 's theories ofhydrodynamics .George Biddell Airy andGeorge Gabriel Stokes had difficulty accepting Russell's experimental observations because they could not be explained by linear water wave theory. His contemporaries spent some time attempting to extend the theory but it would take until1895 beforeDiederik Korteweg andGustav de Vries provided the theoretical explanation. [*cite journal | last = Korteweg | first = D.J. | authorlink = Diederik Korteweg | coauthors =*]Gustav de Vries | title = On the Change of Form of Long Waves advancing in a Rectangular Canal and on a New Type of Long Stationary Waves | journal =Philosophical Magazine | volume = 39 | issue = | pages = pp. 422–443 | publisher = | location = | date = 1895 | url = | format = | issn = | accessdate =(

**Note:**Lord Rayleigh published a paper in Philosophical Magazine in 1876 to support John Scott Russell's experimental observation with his mathematical theory. In his 1876 paper, Lord Rayleigh mentioned Russell's name and also admitted that the first theoretical treatment was by Joseph Valentin Boussinesq in 1871.Joseph Boussinesq mentioned Russell's name in his 1871 paper. Thus Russell's observations on solitons were accepted as true by some prominent scientists within his own life time of 1808-1882. Korteweg and de Vries did not mention John Scott Russell's name at all in their 1895 paper but they did quote Boussinesq's paper in 1871 and Lord Rayleigh's paper in 1876. The paper by Korteweg and de Vries in 1895 was not the first theoretical treatment of this subject but it was a very important milestone in the history of the development of soliton theory.)In 1965

Norman Zabusky ofBell Labs andMartin Kruskal ofPrinceton University first demonstrated soliton behaviour in media subject to theKorteweg–de Vries equation (KdV equation) in a computational investigation using afinite difference approach.In 1967, Gardner, Greene, Kruskal and Miura discovered an

inverse scattering transform enabling analytical solution of the KdV equation. The work ofPeter Lax onLax pair s and the Lax equation has since extended this to solution of many related soliton-generating systems.**olitons in fiber optics**"See also

Soliton (optics) "Much experimentation has been done using solitons in fiber optics applications. Solitons' inherent stability make long-distance transmission possible without the use of

repeater s, and could potentially double transmission capacity as well. [*" [*]*http://www.eetimes.com/showArticle.jhtml?articleID=172302644 Photons advance on two fronts*] ", "EETimes.com", October 24, 2005.In 1973, Akira Hasegawa of

AT&T Bell Labs was the first to suggest that solitons could exist inoptical fiber s, due to a balance betweenself-phase modulation and anomalous dispersion. Also in1973 Robin Bullough made the first mathematical report of the existence of optical solitons. He also proposed the idea of a soliton-based transmission system to increase performance of opticaltelecommunication s.Solitons in a fiber optic system are described by the

Manakov equations .In 1987, P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy, from the Universities of Brussels and Limoges, made the first experimental observation of the propagation of a

dark soliton , in an optical fiber.In 1988, Linn Mollenauer and his team transmitted soliton pulses over 4,000 kilometers using a phenomenon called the

Raman effect , named for the Indian scientist Sir C. V. Raman who first described it in the1920s , to provideoptical gain in the fiber.In 1991, a Bell Labs research team transmitted solitons error-free at 2.5 gigabits per second over more than 14,000 kilometers, using

erbium optical fiber amplifiers (spliced-in segments of optical fiber containing the rare earth element erbium). Pump lasers, coupled to the optical amplifiers, activate the erbium, which energizes the light pulses.In 1998, Thierry Georges and his team at

France Telecom R&D Center, combining optical solitons of different wavelengths (wavelength division multiplexing ), demonstrated a data transmission of 1 terabit per second (1,000,000,000,000 units of information per second).For some reasons, it is possible to observe both positive and negative solitons in optic fibre. However, usually only positive solitons are observed for water waves.(what is the meaning of positive and negative solitons ???)

**olitons in magnets**In magnets, there also exist different types solitons and other nonlinear waves. These magnetic solitons are an exact solution of classical nonlinear differential equations - magnetic equations, e.g. the

Landau-Lifshitz equation , continuumHeisenberg model ,Ishimori equation ,Mikhailov-Yaremchuk equation ,nonlinear Schrodinger equation and so on.**Bions**The bound state of two solitons is known as a "bion".

In field theory Bion usually refers to the solution of the

Born-Infeld model . The name appears to have been coined by G.W.Gibbons in order to distinguish this solution from the conventional soliton, understood as a "regular", finite-energy (and usually stable) solution of a differential equation describing some physical system. The word "regular" means a smooth solution carrying no sources at all. However, the solution of the Born-Infeld model still carries a source in the form of a Dirac-delta function at the origin. As a consequence it displays a singularity in this point (although the electric field is everywhere regular). In some physical contexts (for instance string theory) this feature can be important, which motivated the introduction of a special name for this class of solitons.On the other hand, when gravity is added (i.e. when considering the coupling of the Born-Infeld model to General Relativity) the corresponding solution is called "EBIon", where "E" stands for "Einstein".

**ee also**

*compacton , a soliton with compact support

*Clapotis

*Freak wave s may be a related phenomenon.

*Oscillon s

*peakon , a soliton with a non-differentiable peak.

*Q-ball a non-topological soliton

*Soliton (topological) .

*Soliton (optics)

*Soliton model of nerve impulse propagation

*Spatial soliton

*Solitary wave s in discrete media [*http://www.livescience.com/technology/050614_baby_waves.html*]

*Topological quantum number

*vector soliton **References*** N. J. Zabusky and M. D. Kruskal (1965). "Interaction of 'Solitons' in a Collisionless Plasma and the Recurrence of Initial States." Phys Rev Lett 15, 240

* A. Hasegawa and F. Tappert (1973). "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion." Appl. Phys. Lett. Volume 23, Issue 3, pp. 142-144.

* P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy (1987) "Picosecond steps and dark pulses through nonlinear single mode fibers." Optics. Comm. 62, 374

* P. G. Drazin and R. S. Johnson (1989). "Solitons: an introduction." Cambridge University Press.

* N. Manton and P. Sutcliffe (2004). "Topological solitons." Cambridge University Press.

* Linn F. Mollenauer and James P. Gordon (2006). "Solitons in optical fibers." Elsevier Academic Press.

* R. Rajaraman (1982). "Solitons and instantons." North-Holland.

**External links*** [

*http://www.sciencedaily.com/releases/2005/05/050506141331.htm Solitons, solitary waves and secondary or baby solitary waves in discrete media*]

* [*http://www.ma.hw.ac.uk/solitons/ Heriot-Watt University soliton page*]

* [*http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/index-e.html The many faces of solitons*]

* [*http://www.usf.uni-osnabrueck.de/~kbrauer/solitons.html Klaus Brauer's soliton page*]

* [*http://homepages.tversu.ru/~s000154/collision/main.html Solitons and Soliton Collisions*]

* [*http://www.ma.hw.ac.uk/~chris/scott_russell.html John Scott Russell and the solitary wave*]

* [*http://www.severn-bore.co.uk/default.htm Severn Bore web site*]

* [*http://www.ma.hw.ac.uk/solitons/LocalHeroes/sr.html John Scott Russell biography*]

* [*http://unic.ece.cornell.edu Soliton in Electrical Engineering*]

* [*http://web.njit.edu/~miura/ Miura's home page*]

* [*http://www.ma.hw.ac.uk/solitons/soliton1b.html Photograph of Soliton on the Scott Russell Aqueduct*]

* [*http://www.pnas.org/cgi/reprint/102/28/9790 Solitons possible agent of nerve transmission (PDF)(pnas.org)*]

* [*http://www.youtube.com/watch?v=H4rN3Wr4ctw Three Solitons Solution of KdV Equation*]

* [*http://www.youtube.com/watch?v=5z5SylS2QHE Three Solitons (unstable) Solution of KdV Equation*]

* [*http://lie.math.brocku.ca/~sanco/solitons/index.html Solitons & Nonlinear Wave Equations*]

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