Solitons

In General
* Idea: Smooth, particle-like (i.e., localized, finite-energy, and stable under scattering) solutions of non-linear field theories, usually described by a finite number of parameters; They are either absolute minima of the energy, or minima in some sector determined by other conserved quantities.
* Analog: Runners on a soft mattress; Faster ones run uphill, slower ones downhill, and the group doesn't spread.
* History: The term was introduced in the study of properties of some solutions of the Korteweg-de Vries equation in collision, as shown by computer simulations.

Observation and Phenomenology > s.a. sources of gravitational radiation.
* History: The first reported one occurred in the 1880's in the water of a narrow channel; Other solitary waves have been seen more recently in ocean water.
* Soliton stars: Can be formed even with simple models of a free scalar field and gravity, from simple gravitational collapse, but there is a whole zoo of them.
@ General references: Osborne & Burch Sci(80)may [nice historical and theoretical setting].
@ And fractional charges: Jackiw & Rebbi PRD(76); Jackiw & Schrieffer NPB(81).

Properties, Techniques
* Scattering and radiation: In 1+1 dimensions, scattering is radiationless, but not in higher dimensions; Therefore, if there is any interaction, we expect bremsstrahlung and no (evolution) invariant pure solitonic subspace of phase space (except the 1-soliton sector and plane waves).
* Bäcklund transformation: In gauge theories, a transformation of the fields that adds (or subtracts) a soliton.
@ Bäcklund transformation: Kuznetsov & Vanhaecke JGP(02)nl/00 [geometric]; > s.a. integrable systems.

Examples > s.a. Sine-Gordon and Korteweg-de Vries Equation; monopoles; non-commutative field theory; Non-Linear Quantum Mechanics.
* Topological: Solitonic solutions of the KdV equation (see above); Kinks of the Sine-Gordon theory; Monopoles in non-abelian gauge theories; Monopoles in Kaluza-Klein theory or black holes with charges in gr (in the latter case we have to give up smoothness or some other condition –like flat topology– because of the Lichnerowicz theorem).
* Non-topological: Q-Balls (& Coleman).
@ Supersymmetric theories: Aichelburg & Embacher PRD(88), PRD(88), PRD(88), PRD(88) [sugra]; Eto et al ht/05-in [supersymmetric gauge theories]; Tong ht/05-ln [supersymmetric gauge theories and string theory]; Shifman & Young RMP(07) [critical solitons, rev].
@ Yang-Mills theory, topological: Friedman & Sorkin CMP(83), CMP(83); Alonso et al ht/06-ln [masses]; > s.a. QCD phenomenology.
@ (Non-linear) Dirac theory: Finkelstein PR(51); Soler PRD(70); Grosse PRP(86).
@ Maxwell-Dirac theory: Bohun & Cooperstock PRA(99)phy/00; > s.a. dirac fields in curved spacetime.
@ Einstein-Yang-Mills theory: Maison gq/96 [rev]; Corichi et al PRD(01)gq [mass formula]; Gal'tsov ht/01-GR16; Oliynyk & Künzle CQG(02)gq/01 [spherical]; Gal'tsov & Davydov G&C(06), PRD(07) [cylindrically symmetric]; Hod PLB(07)-a0711.
@ General relativity: Morris & Dodd ed; Cadavid & Finkelstein PRD(98) [non-linear scalar field]; Saha gq/98/G&C [from scalar and electromagnetic field]; Galloway et al PRL(02)ht/01 [AdS, uniqueness]; Belinski & Verdaguer 01; Cebeci et al PRD(06) [in D-dimensional AdS, M < 0].
@ 5D general relativity: Sajko & Wesson MPLA(01) [energy]; Azuma & Koikawa PTP(06)ht/05; > s.a. kaluza-klein phenomenology and solutions.
@ O(3) sigma model: Govindarajan MPLA(98) [knot solitons]; Battye & Sutcliffe PRS(99)ht/98.
@ Chern-Simons theory: Chung et al AP(01) [self-dual]; Kim LMP(02)mp/01 [self-dual, on a cylinder].
@ Knotted solitons: Niemi PRD(00)ht/99; Finkelstein ht/07 [electroweak theory + SU₂(2)]
@ In fuzzy physics: Baez et al CMP(00)ht/98.
@ Related topics: Goldstone & Wilczek PRL(81) [fractional quantum numbers]; Davis PRD(88) [semi-topological]; Youm NPB(00), NPB(00), NPB(00) [brane world]; Radu gq/05-in [rotating]; Ferreira JHEP(06) [t-dependent Hopf solitons]; > s.a. Lamé Equation.

References
@ Simple: Herman AS(92).
@ History: Zabusky & Kruskal PRL(65); Allen PS(98) [early].
@ Books and reviews: Faddeev & Korepin PRP(78) [quantum theory]; Bullough & Caudrey ed-80; Eilenberger 81; in Felsager 81; Dodd et al 82; Rajaraman 82; Rebbi & Soliani ed-84; PTPS(88)#94; Ward ht/05-in.
@ Geometry and topology: Riazi ht/01/IJTGN.
@ Related topics: Olive CzJP(79) [supermetric]; Palais BAMS(97) [symmetries, history]; Rehren LMP(98) [spin and statistics].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 27 jun 2008