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{{Short description|Type of nonlinear wave in physics}}
{{about||the component in an internal combustion engine|Crankcase ventilation system|the item of scuba apparatus|rebreather|other uses}}

In physics, a '''breather''' is a [[nonlinear]] [[wave]]  in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for [[infinitesimal]] [[amplitude]]s, which tends towards an even distribution of initially localized energy.
A '''discrete breather''' is a breather solution on a nonlinear [[lattice model (physics)|lattice]].

The term breather originates from the characteristic that most breathers are localized in space and oscillate (as a [[breath]]e) in time.<ref name=AKNS73>{{cite journal| title = Method for solving the sine-Gordon equation | author = M. J. Ablowitz |author2= D. J. Kaup |author3= A. C. Newell |author4= H. Segur | year = 1973 | journal = Physical Review Letters | volume = 30 |       pages = 1262–1264 | doi = 10.1103/PhysRevLett.30.1262 | issue = 25 | bibcode=1973PhRvL..30.1262A}}</ref> But also the opposite situation: oscillations in space and localized in time{{Clarify|date=May 2012}}, is denoted as a breather.

[[File:Breather-surface.png|right|thumb|300px|This breather pseudospherical surface corresponds to a solution  of a non-linear wave-equation.]]
[[File:Breather-4-11.png|right|thumb|250px|Pseudospherical breather surface]]

==Overview==
[[Image:Sine gordon 5.gif|frame|''[[Sine-Gordon]] standing breather'' is a swinging in time coupled kink-antikink 2-soliton solution.]]
[[Image:Sine gordon 6.gif|frame|''Large amplitude moving [[sine-Gordon]] breather''.]]
A breather is a localized [[periodic function|period]]ic solution of either [[continuum mechanics|continuous media]] equations or discrete [[lattice (group)|lattice]] equations. The exactly solvable [[sine-Gordon equation]]<ref name=AKNS73/> and the focusing [[nonlinear Schrödinger equation]]<ref name=AEK87>{{cite journal | title = First-order exact solutions of the nonlinear Schrödinger equation | author = N. N. Akhmediev |author2= V. M. Eleonskiǐ|author3= N. E. Kulagin | year = 1987 | journal = Theoretical and Mathematical Physics | volume = 72 | pages = 809–818 | doi = 10.1007/BF01017105|bibcode = 1987TMP....72..809A | issue = 2 | s2cid = 18571794 }} Translated from ''Teoreticheskaya i Matematicheskaya Fizika'' 72(2): 183–196, August, 1987.</ref> are examples of one-[[dimensional]] [[partial differential equation]]s that possess breather solutions.<ref>{{cite book| title = Solitons, non-linear pulses and beams | author = N. N. Akhmediev |author2= A. Ankiewicz | year = 1997 | publisher = Springer | isbn = 978-0-412-75450-0}}</ref> Discrete nonlinear [[Hamiltonian lattices]] in many cases support breather solutions.

Breathers are [[soliton]]ic structures. There are two types of breathers: [[standing wave|standing]] or [[traveling wave|traveling]] ones.<ref>Miroshnichenko A, Vasiliev A, Dmitriev S. ''[http://homepages.tversu.ru/~s000154/collision/main.html Solitons and Soliton Collisions]''.</ref> Standing breathers correspond to localized solutions whose amplitude vary in time (they are sometimes called [[oscillon]]s). A necessary condition for the existence of breathers in discrete lattices is that the breather main [[frequency]] and all its multipliers are located outside of the [[phonon]] [[spectrum]] of the lattice.

==Example of a breather solution for the sine-Gordon equation==
The [[sine-Gordon equation]] is the nonlinear [[dispersive partial differential equation]]

:<math>\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0,</math>

with the [[field (physics)|field]] ''u'' a function of the spatial coordinate ''x'' and time ''t''.

An exact solution found by using the [[inverse scattering transform]] is:<ref name=AKNS73/>

:<math>u = 4 \arctan\left(\frac{\sqrt{1-\omega^2}\;\cos(\omega t)}{\omega\;\cosh(\sqrt{1-\omega^2}\; x)}\right),</math>

which, for ''ω < 1'', is periodic in time ''t'' and [[exponential decay|decays exponentially]] when moving away from ''x = 0''.

==Example of a breather solution for the nonlinear Schrödinger equation==
The focusing [[nonlinear Schrödinger equation]]<ref>The focusing [[nonlinear Schrödinger equation]] has a nonlinearity parameter ''κ'' of the same [[sign (mathematics)]] as the dispersive term proportional to ''&part;<sup>2</sup>u/&part;x<sup>2</sup>'', and has [[soliton]] solutions. In the de-focusing [[nonlinear Schrödinger equation]] the nonlinearity parameter is of opposite sign.</ref> is the dispersive partial differential equation:

:<math>i\,\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2} + |u|^2 u = 0,</math>

with ''u'' a [[complex number|complex]] field as a function of ''x'' and ''t''. Further ''i'' denotes the [[imaginary unit]].

One of the breather solutions (Kuznetsov-Ma breather) is <ref name=AEK87/>

:<math>
  u =
  \left(
    \frac{2 b^2 \cosh(\theta) + 2 i b \sqrt{2-b^2} \sinh(\theta)}
         {2 \cosh(\theta)-\sqrt{2}\sqrt{2-b^2} \cos(a b x)} - 1
  \right)a\,e^{i a^2 t}
</math>

with

:<math>
  \theta=a^2\,b\,\sqrt{2-b^2}\;t,
</math>

which gives breathers periodic in space ''x'' and approaching the uniform value ''a'' when moving away from the focus time ''t'' = 0. These breathers exist for values of the [[modulation]] parameter ''b'' less than {{radic|2}}.
Note that a limiting case of the breather solution is the [[Peregrine soliton]].<ref>{{cite journal |last1= Kibler|first1=B. |last2= Fatome |first2= J. | last3= Finot |first3= C. | last4= Millot |first4= G. | last5= Dias |first5= F. | last6= Genty |first6= G. | last7= Akhmediev |first7= N. | last8= Dudley |first8= J.M. | year= 2010 |title= The Peregrine soliton in nonlinear fibre optics|journal= Nature Physics|doi= 10.1038/nphys1740 | volume=6 | issue=10|bibcode = 2010NatPh...6..790K |pages= 790 |doi-access= free }}</ref>

==See also==
* [[Breather surface]]
* [[Soliton]]

==References and notes==
{{Reflist}}

[[Category:Waves]]