Editing Soliton (section)

==History==
[[File:JohnScottRussellPlaque.png|thumb|A plaque marking the workshop of John Scott Russell at 8 Stafford Street in [[Edinburgh]]]]
In 1834, [[John Scott Russell]] described his ''[[wave of translation]]'':<ref group=nb>"Translation" here means that there is real mass transport, although it is not the same water which is transported from one end of the canal to the other end by this "Wave of Translation". Rather, a [[fluid parcel]] acquires [[momentum]] during the passage of the solitary wave, and comes to rest again after the passage of the wave. But the fluid parcel has been displaced substantially forward during the process&nbsp;– by [[Stokes drift]] in the wave propagation direction. And a net mass transport is the result. Usually there is little mass transport from one side to another side for ordinary waves.</ref><ref group=nb>This passage has been repeated in many papers and books on soliton theory.</ref>

{{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&nbsp;– 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.<ref>{{cite book |first=J. |last=Scott Russell |author-link=John Scott Russell |title=Report on Waves: Made to the Meetings of the British Association in 1842–43 |url=https://books.google.com/books?id=994EAAAAYAAJ&pg=PA1 |year=1845 }}</ref>}}

Scott 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&nbsp;– 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.

Scott Russell's experimental work seemed at odds with [[Isaac Newton]]'s and [[Daniel Bernoulli]]'s theories of [[hydrodynamics]]. [[George Biddell Airy]] and [[George Gabriel Stokes]] had difficulty accepting Scott Russell's experimental observations because they could not be explained by the existing water wave theories. Additional observations were reported by [[Henri-Émile Bazin|Henry Bazin]] in 1862 after experiments carried out in the [[canal de Bourgogne]] in France.<ref>{{Cite journal |last=Bazin |first=Henry |author-link=Henri-Émile Bazin |year=1862 |title=Expériences sur les ondes et la propagation des remous |journal=Comptes Rendus des Séances de l'Académie des Sciences |language=french |volume=55 |pages=353–357}}</ref> Their contemporaries spent some time attempting to extend the theory but it would take until the 1870s before [[Joseph Boussinesq]]<ref>{{cite journal |last=Boussinesq |first=J. |title=Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire |journal=[[Comptes Rendus de l'Académie des Sciences|C. R. Acad. Sci. Paris]] |volume=72 |year=1871}}</ref> and [[Lord Rayleigh]] published a theoretical treatment and solutions.<ref group=nb>[[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 Scott 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 Scott Russell's observations on solitons were accepted as true by some prominent scientists within his own lifetime of 1808–1882.</ref> In 1895 [[Diederik Korteweg]] and [[Gustav de Vries]] provided what is now known as the [[Korteweg–de Vries equation]], including solitary wave and periodic [[cnoidal wave]] solutions.<ref>{{cite journal | last1 = Korteweg  | first1 = D. J.  | author-link1 = Diederik Korteweg  | first2=G. | last2=de Vries | author-link2=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 = 240  | pages = 422–443  | year = 1895 | doi=10.1080/14786449508620739| url = https://zenodo.org/record/1431215 }}</ref><ref group=nb>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 of 1871 and Lord Rayleigh's paper of 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.</ref>

[[File:BBM equation - overtaking solitary waves animation.gif|thumb|416px|right|
An animation of the overtaking of two solitary waves according to the [[Benjamin–Bona–Mahony equation]] – or BBM equation, a model equation for (among others) long [[surface gravity wave]]s. The [[wave height]]s of the solitary waves are 1.2 and 0.6, respectively, and their velocities are 1.4 and 1.2.{{paragraph}}
The ''upper graph'' is for a [[frame of reference]] moving with the average velocity of the solitary waves.{{paragraph}}
The ''lower graph'' (with a different vertical scale and in a stationary frame of reference) shows the [[oscillation|oscillatory]] tail produced by the interaction.<ref>{{Cite journal | doi = 10.1063/1.863011 | volume = 23 | issue = 3 | pages = 438–441 | last1 = Bona | first1 = J. L. | author1-link=Jerry L. Bona | first2 = W. G. | last2 = Pritchard | first3 = L. R. |last3 = Scott | title = Solitary-wave interaction | journal = Physics of Fluids | year = 1980 |bibcode = 1980PhFl...23..438B }}</ref> Thus, the solitary wave solutions of the BBM equation are not solitons.]]
In 1965 [[Norman Zabusky]] of [[Bell Labs]] and [[Martin Kruskal]] of [[Princeton University]] first demonstrated soliton behavior in media subject to the [[Korteweg–de Vries equation]] (KdV equation) in a computational investigation using a [[finite difference]] approach.  They also showed how this behavior explained the puzzling earlier work of [[Fermi–Pasta–Ulam–Tsingou problem|Fermi, Pasta, Ulam, and Tsingou]].<ref name="Zabusky 1965"/>

In 1967, Gardner, Greene, Kruskal and Miura discovered an [[inverse scattering transform]] enabling [[analytic function|analytical]] solution of the KdV equation.<ref>{{Cite journal
| doi = 10.1103/PhysRevLett.19.1095
| volume = 19
| issue = 19
| pages = 1095–1097
| last1 = Gardner
| first1 = Clifford S.
| first2 = John M.
| last2 = Greene
| first3 = Martin D.
| last3 = Kruskal
| first4 = Robert M.
| last4 = Miura
| title = Method for Solving the Korteweg–deVries Equation
| journal = Physical Review Letters
| year = 1967
| bibcode=1967PhRvL..19.1095G
}}</ref> The work of [[Peter Lax]] on [[Lax pair]]s and the Lax equation has since extended this to solution of many related soliton-generating systems.

Solitons are, by definition, unaltered in shape and speed by a collision with other solitons.<ref>{{Cite book | publisher = Springer | isbn = 9783540659198 | last = Remoissenet | first = M. | title = Waves called solitons: Concepts and experiments | year = 1999 | page = [https://archive.org/details/wavescalledsolit0000remo/page/11 11] | url = https://archive.org/details/wavescalledsolit0000remo/page/11 }}</ref> So solitary waves on a water surface are ''near''-solitons, but not exactly – after the interaction of two (colliding or overtaking) solitary waves, they have changed a bit in [[amplitude]] and an oscillatory residual is left behind.<ref>See e.g.: <br>• {{Cite journal | doi = 10.1017/S0022112076003194 | volume = 76 | issue = 1 | pages = 177–186 | last = Maxworthy | first = T. | title = Experiments on collisions between solitary waves | journal = Journal of Fluid Mechanics | year = 1976 |bibcode = 1976JFM....76..177M | s2cid = 122969046 }}<br>• {{Cite journal | doi = 10.1017/S0022112082001141 | volume = 118 | pages = 411–443 | last1 = Fenton | first1 = J.D. | first2 = M.M. | last2 = Rienecker | title = A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions | journal = Journal of Fluid Mechanics | year = 1982 | doi-broken-date = 29 November 2024 |bibcode = 1982JFM...118..411F | s2cid = 120467035 }}<br>• {{Cite journal | doi = 10.1063/1.2205916 | volume = 18 | issue = 57106 | pages = 057106–057106–25 | last1 = Craig | first1 = W. | first2 = P. | last2 = Guyenne |first3 = J. | last3 = Hammack | first4= D. | last4 = Henderson |first5 = C. | last5 = Sulem | title = Solitary water wave interactions | journal = Physics of Fluids | year = 2006 |bibcode = 2006PhFl...18e7106C }}</ref>

Solitons are also studied in quantum mechanics, thanks to the fact that they could provide a new foundation of it through [[de Broglie]]'s unfinished program, known as "Double solution theory" or "Nonlinear wave mechanics". This theory, developed by de Broglie in 1927 and revived in the 1950s, is the natural continuation of his ideas developed between 1923 and 1926, which extended the [[wave–particle duality]] introduced by [[Albert Einstein]] for the [[light quanta]], to all the particles of matter. The observation of accelerating surface gravity water wave soliton using an external hydrodynamic linear potential was demonstrated in 2019. This experiment also demonstrated the ability to excite and measure the phases of ballistic solitons.<ref>{{Cite journal | doi = 10.1103/PhysRevE.101.050201
 | volume = 101 | issue = 5 | last = G. G. Rozenman | first =  A. Arie, L. Shemer| title = Observation of accelerating solitary wavepackets | journal = Phys. Rev. E  | year = 2019 | page = 050201 | pmid = 32575227 | s2cid = 219506298 }}</ref>