Solitons  

In General > s.a. Bäcklund Transformations.
* 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.
> Online resources: see Wikipedia page.

Observation and Phenomenology > s.a. particle models; sources of gravitational radiation.
* History: The first reported one occurred in the 1880s in the water of a narrow channel, by John Scott Russell is his studies of efficient hull forms for ships; Other solitary waves have been seen more recently in ocean water.
* 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).
* 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].
@ In water: Ablowitz & Baldwin PRE(12) [in shallow-water waves]; Chabchoub et al PRL(13) [observation of dark solitons].
@ And fractional charges: Jackiw & Rebbi PRD(76); Jackiw & Schrieffer NPB(81).

Examples > s.a. Faddeev Model; Sine-Gordon and Korteweg-de Vries Equation; monopoles; Non-Linear Quantum Mechanics; topological defects.
* Topological: Solutions of a field theory that owe their existence to a multiplicity of ground states, forming a space with non-trivial topology; Solitonic solutions of the KdV equation (see above); Kinks of the Sine-Gordon theory; Monopoles and other topological defects in non-abelian gauge theories; Monopoles in Kaluza-Klein theory or black holes with charges in general relativity (in the latter case we have to give up smoothness or some other condition –like flat topology– because of the Lichnerowicz theorem); Vortices in a type-II superconductor, forming an Abrikosov lattice.
* Non-topological: Solutions of a field theory that owe their existence to the non-linearity of the field equations; e.g., Q-Balls (& Coleman).
@ Supersymmetric theories: Aichelburg & Embacher PRD(88), PRD(88), PRD(88), PRD(88) [supergravity]; Shifman LNP(05); Eto et al AIP(05)ht [supersymmetric gauge theories]; Tong ht/05-ln [supersymmetric gauge theories and string theory]; Shifman & Young RMP(07) [critical solitons, rev]; Shifman & Yung 09.
@ 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]; Sajko & Wesson MPLA(01) [5D, energy]; Galloway et al PRL(02)ht/01 [AdS, uniqueness]; Belinski & Verdaguer 01; Cebeci et al PRD(06) [d-dimensional AdS, M < 0]; Azuma & Koikawa PTP(06)ht/05 [5D]; > 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].
@ Non-topological: Smolyakov JMP(11)-a1012 [Yang-Mills theory coupled to a non-linear scalar field, conditions for existence].
@ Knotted solitons: Niemi PRD(00)ht/99; Finkelstein ht/07 [electroweak theory + SU2(2)].
@ In generalized theories: Baez et al CMP(00)ht/98 [fuzzy physics]; > s.a. non-commutative gauge theories.
@ Related topics: Su et al PRL(79) [topological, in polyacetylene]; Goldstone & Wilczek PRL(81) [fractional quantum numbers]; Davis PRD(88) [semi-topological]; Youm NPB(00), NPB(00), NPB(00) [brane world]; Radu gq/05-conf [rotating]; Ferreira JHEP(06) [time-dependent Hopf solitons]; Endlich et al JHEP(11)-a1002 [no-stable-soliton result for theories with derivative interactions]; Ding PLA(10) [in KdV equation, and motion of spacelike curves in 3D Minkowski space]; Zamboni-Rached & Recami JMP(12) [solutions of the Schrödinger equation]; Beccaria et al TMP(12) [in 1+1 and 2+1 dimensions]; > s.a. Lamé Equation.

References
@ Simple: Herman AS(92).
@ History: Zabusky & Kruskal PRL(65); Allen PS(98) [early].
@ Books and reviews: 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-en; Weinberg 12.
@ Quantum theory: Faddeev & Korepin PRP(78); Dvali et al a1508 [coherent state picture].
@ Geometry and topology: Riazi ht/01/IJTGN.
@ Related topics: Olive CzJP(79) [supermetric]; Palais BAMS(97) [symmetries, history]; Rehren LMP(98) [spin and statistics]; Tao BAMS(09) [stability]; Al-Alawi a0911 [dynamics with potential barriers]; Weiner IJTP(10) [fermions are topological solitons]; Kuznetsov & Dias PRP(11) [bifurcations and stability].


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