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 + SU_{2}(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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