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| The '''Sine–Gordon equation''' is a nonlinear hyperbolic [[partial differential equation]] in 1 + 1 dimensions involving the [[d'Alembert operator]] and the [[sine function|sine]] of the unknown function. It was originally introduced by {{harvs|txt|first=Edmond|last=Bour|authorlink=Edmond Bour|year=1862}} in the course of study of [[pseudosphere|surfaces of constant negative curvature]] as the [[Gauss–Codazzi equation]] for surfaces of curvature –1 in 3-space, and rediscovered by {{harvs|txt|last1=Frenkel|last2= Kontorova|year=1939}} in their study of crystal dislocations. This equation attracted a lot of attention in the 1970s due to the presence of [[soliton]] solutions.
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| == Origin of the equation and its name ==
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| There are two equivalent forms of the sine–Gordon equation. In the ([[real number|real]]) ''space-time coordinates'', denoted (''x'', ''t''), the equation reads:<ref name="Rajaraman1989">{{cite book
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| |last = Rajaraman
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| |first = R.
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| | title = Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory
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| | publisher = North-Holland
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| | series = North-Holland Personal Library
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| | volume = 15
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| | pages = 34–45
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| | year = 1989
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| | doi =
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| | isbn = 978-0-444-87047-6
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| }}</ref>
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| :<math>\, \varphi_{tt}- \varphi_{xx} + \sin\varphi = 0.</math>
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| Passing to the ''light cone coordinates'' (''u'', ''v''), akin to ''asymptotic coordinates'' where
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| : <math> u=\frac{x+t}2, \quad v=\frac{x-t}2, </math>
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| the equation takes the form:<ref name="Polyanin2004">{{cite book
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| |last = Polyanin
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| |first = Andrei D.
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| |coauthors = Valentin F. Zaitsev
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| | title = Handbook of Nonlinear Partial Differential Equations
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| | publisher = Chapman & Hall/CRC Press
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| | series =
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| | pages = 470–492
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| | year = 2004
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| | doi =
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| | isbn = 978-1-58488-355-5
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| }}</ref>
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| :<math>\varphi_{uv} = \sin\varphi.\,</math>
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| This is the original form of the sine–Gordon equation, as it was considered in the nineteenth century in the course of investigation of [[differential geometry of surfaces|surfaces]] of constant [[Gaussian curvature]] ''K'' = −1, also called [[pseudospherical surface]]s. Choose a coordinate system for such a surface in which the coordinate mesh ''u'' = constant, ''v'' = constant is given by the [[asymptotic curve|asymptotic line]]s parameterized with respect to the arc length. The [[first fundamental form]] of the surface in these coordinates has a special form
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| : <math> ds^2 = du^2 + 2\cos\varphi \,du\, dv + dv^2,\, </math>
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| where <math> \varphi </math> expresses the angle between the asymptotic lines, and for the [[second fundamental form]], ''L'' = ''N'' = 0. Then the [[Codazzi-Mainardi equation]] expressing a compatibility condition between the first and second fundamental forms results in the sine–Gordon equation. The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by [[Luigi Bianchi|Bianchi]] and [[Albert Victor Bäcklund|Bäcklund]] led to the discovery of [[Bäcklund transformation]]s.
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| The name "sine–Gordon equation" is a pun on the well-known [[Klein–Gordon equation]] in physics:
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| :<math> \varphi_{tt}- \varphi_{xx} + \varphi\ = 0.\,</math>
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| The sine–Gordon equation is the [[Euler–Lagrange equation]] of the field whose [[Lagrangian density]] is given by
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| :<math>\mathcal{L}_\text{SG}(\varphi) = \frac{1}{2}(\varphi_t^2 - \varphi_x^2) - 1 + \cos\varphi.</math>
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| Using the Taylor series expansion of the [[cosine]] in the Lagrangian,
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| :<math>\cos(\varphi) = \sum_{n=0}^\infty \frac{(-\varphi ^2)^n}{(2n)!},</math>
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| it can be rewritten as the [[Scalar_field_theory#Linear_.28free.29_theory|Klein–Gordon Lagrangian]] plus higher order terms | |
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| :<math>
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| \begin{align}
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| \mathcal{L}_\text{SG}(\varphi) & = \frac{1}{2}(\varphi_t^2 - \varphi_x^2) - \frac{\varphi^2}{2} + \sum_{n=2}^\infty \frac{(-\varphi^2)^n}{(2n)!} \\
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| & = \mathcal{L}_\text{KG}(\varphi) + \sum_{n=2}^\infty \frac{(-\varphi^2)^n}{(2n)!}.
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| \end{align}
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| </math>
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| == Soliton solutions ==
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| An interesting feature of the sine–Gordon equation is the existence of [[soliton]] and multisoliton solutions.
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| === 1-soliton solutions ===
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| The sine–Gordon equation has the following 1-[[soliton]] solutions:
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| :<math>\varphi_\text{soliton}(x, t) := 4 \arctan e^{m \gamma (x - v t) + \delta}\,</math>
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| where
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| : <math>\gamma^2 = \frac{1}{1 - v^2}.</math>
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| and the slightly more general form of the equation is assumed:
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| :<math>\, \varphi_{tt}- \varphi_{xx} + m^2 \sin\varphi = 0.</math>
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| The 1-soliton solution for which we have chosen the positive root for <math>\gamma</math> is called a ''kink'', and represents a twist in the variable <math>\varphi</math> which takes the system from one solution <math>\varphi=0</math> to an adjacent with <math>\varphi=2\pi</math>. The states <math>\varphi=0(\textrm{mod}2\pi)</math> are known as vacuum states as they are constant solutions of zero energy. The 1-soliton solution in which we take the negative root for <math>\gamma</math> is called an ''antikink''. The form of the 1-soliton solutions can be obtained through application of a Bäcklund transform to the trivial (constant vacuum) solution and the integration of the resulting first-order differentials:
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| :<math>{\varphi^\prime}_u = \varphi_u + 2\beta\sin\left(\frac{\varphi^\prime + \varphi}{2}\right),</math>
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| :<math>{\varphi^\prime}_v = -\varphi_v + \frac{2}{\beta} \sin\left(\frac{\varphi^\prime - \varphi}{2}\right)\text{ with }\varphi = \varphi_0 = 0</math>
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| for all time.
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| The 1-soliton solutions can be visualized with the use of the elastic ribbon sine–Gordon model as discussed by ''Dodd and co-workers''.<ref name="Dodd1982">{{cite book
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| |last = Dodd
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| |first = Roger K.
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| | coauthors = J. C. Eilbeck, J. D. Gibbon, H. C. Morris
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| | title = Solitons and Nonlinear Wave Equations
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| | publisher = Academic Press
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| | location = London
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| | series =
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| | volume =
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| | pages =
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| | year = 1982
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| | doi =
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| | isbn = 978-0-12-219122-0
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| }}</ref> Here we take a clockwise ([[Right-hand rule|left-handed]]) twist of the elastic ribbon to be a kink with topological charge <math>\vartheta_{\textrm{K}}=-1</math>. The alternative counterclockwise ([[Right-hand rule|right-handed]]) twist with topological charge <math>\vartheta_{\textrm{AK}}=+1</math> will be an antikink.
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| {|
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| |-
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| | [[Image:Sine gordon 1.gif|frame|Traveling ''kink'' soliton represents propagating clockwise twist.<ref name="Georgiev2004">{{cite journal
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| | author = Georgiev DD, Papaioanou SN, Glazebrook JF
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| | title = Neuronic system inside neurons: molecular biology and biophysics of neuronal microtubules
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| | journal = Biomedical Reviews
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| | volume = 15
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| | issue =
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| | pages = 67–75
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| | year = 2004
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| | url = http://cogprints.org/4364/
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| | doi =
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| }}</ref><ref name="Georgiev2007">{{cite journal
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| | author = Georgiev DD, Papaioanou SN, Glazebrook JF
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| | title = Solitonic effects of the local electromagnetic field on neuronal microtubules
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| | journal = Neuroquantology
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| | volume = 5
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| | issue = 3
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| | pages = 276–291
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| | year = 2007
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| | url = http://cogprints.org/3894/
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| | doi =
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| }}</ref> ]]
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| | [[Image:Sine gordon 2.gif|frame|Traveling ''antikink'' soliton represents propagating counterclockwise twist.<ref name="Georgiev2004"/><ref name="Georgiev2007"/>]]
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| |}
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| === 2-soliton solutions ===
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| Multi-[[soliton]] solutions can be obtained through continued application of the [[Bäcklund transform]] to the 1-soliton solution, as prescribed by a [[Bianchi lattice]] relating the transformed results.<ref name="Rogers2002">{{cite book
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| |last = Rogers
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| |first = C.
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| | coauthors = W. K. Schief
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| | title = Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
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| | publisher = [[Cambridge University Press]]
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| | location = New York
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| | series = Cambridge Texts in Applied Mathematics
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| | volume =
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| | pages =
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| | year = 2002
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| | doi =
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| | isbn = 978-0-521-01288-1
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| }}</ref> The 2-soliton solutions of the sine–Gordon equation show some of the characteristic features of the solitons. The traveling sine–Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a [[Phase (waves)|phase shift]]. Since the colliding solitons recover their [[velocity]] and [[shape]] such kind of [[interaction]] is called an [[elastic collision]].
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| {|
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| | [[Image:Sine gordon 3.gif|frame|''Antikink-kink'' collision.<ref name="Georgiev2004"/><ref name="Georgiev2007"/>]]
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| | [[Image:Sine gordon 4.gif|frame|''Kink-kink'' collision.<ref name="Georgiev2004"/><ref name="Georgiev2007"/>]]
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| |}
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| Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a ''[[breather]]''. There are known three types of breathers: ''standing breather'', ''traveling large amplitude breather'', and ''traveling small amplitude breather''.<ref name = "mir">Miroshnichenko A, Vasiliev A, Dmitriev S. ''[http://homepages.tversu.ru/~s000154/collision/main.html Solitons and Soliton Collisions].''</ref>
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| {|
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| | [[Image:Sine gordon 5.gif|frame|''Standing breather'' is a swinging in time coupled kink-antikink soliton.<ref name="Georgiev2004"/><ref name="Georgiev2007"/>]]
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| | [[Image:Sine gordon 6.gif|frame|''Large amplitude moving breather''.<ref name="Georgiev2004"/><ref name="Georgiev2007"/>]]
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| |}
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| {|
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| | [[Image:Sine gordon 7.gif|frame|''Small-amplitude moving breather'' — looks exotic but essentially has a breather envelope.<ref name="Georgiev2004"/><ref name="Georgiev2007"/>]]
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| |}
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| === 3-soliton solutions ===
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| 3-soliton collisions between a traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather,
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| the shift of the breather <math>\Delta_{\textrm{B}}</math> is given by:
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| : <math>\Delta_B =\frac{2\textrm{arctanh}\sqrt{(1-\omega^{2})(1-v_\text{K}^2)}}{\sqrt{1-\omega^{2}}}</math>
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| where <math>v_\text{K}</math> is the velocity of the kink, and <math>\omega</math> is the breather's frequency.<ref name = "mir"/> If the old position of the standing breather is <math>x_{0}</math>, after the collision the new position will be <math>x_0 + \Delta_\text{B}</math>.
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| {|
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| | [[Image:Sine gordon 8.gif|frame|''Moving kink-standing breather'' collision.<ref name="Georgiev2004"/><ref name="Georgiev2007"/>]]
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| | [[Image:Sine gordon 9.gif|frame|''Moving antikink-standing breather'' collision.<ref name="Georgiev2004"/><ref name="Georgiev2007"/>]]
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| |}
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| ==Related equations==
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| The '''{{visible anchor|sinh–Gordon equation}}''' is given by
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| :<math>\varphi_{xx}- \varphi_{tt} = \sinh\varphi.\,</math>
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| This is the [[Euler–Lagrange equation]] of the [[Lagrangian]]
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| :<math>\mathcal{L}={1\over 2}(\varphi_t^2 - \varphi_x^2) - \cosh\varphi.\,</math>
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| Another closely related equation is the '''elliptic sine–Gordon equation''', given by
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| :<math>\varphi_{xx} + \varphi_{yy} = \sin\varphi,\,</math>
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| where <math>\varphi</math> is now a function of the variables ''x'' and ''y''. This is no longer a soliton equation, but it has many similar properties, as it is related to the sine–Gordon equation by the [[analytic continuation]] (or [[Wick rotation]]) ''y'' = i''t''.
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| The '''elliptic sinh–Gordon equation''' may be defined in a similar way.
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| A generalization is given by [[Toda field theory]].
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| ==Quantum version==
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| In quantum field theory the sine–Gordon model contains a parameter, it can be identified with [[Planck constant]]. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of breathers. The number of the breathers depends on the value of the parameter. Multi particle productions cancels on mass shell. Vanishing of two into four amplitude was explicitly checked in one loop approximation.
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| Semi-classical quantization of the sine–Gordon model was done by [[Ludwig Faddeev]] and [[Vladimir Korepin]].<ref name="Faddeev1978">{{cite journal
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| | author = Faddeev LD, Korepin VE
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| | title = Quantum theory of solitons
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| | journal = Physics Reports
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| | volume = 42
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| | issue = 1
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| | pages = 1–87
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| | year = 1978
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| | pmid =
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| | doi = 10.1016/0370-1573(78)90058-3
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| |bibcode = 1978PhR....42....1F }}</ref> The exact quantum scattering matrix was discovered by [[Alexander Zamolodchikov]].
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| This model is [[S-duality|S-dual]] to the [[Thirring model]].
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| ==In finite volume and on a half line==
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| One can also consider the sine–Gordon model on a circle, on a line segment, or on a half line. It is possible to find boundary conditions which preserve the integrability of the model. On a half line the spectrum contains [[boundary bound state]]s in addition to the solitons and breathers.
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| ==Supersymmetric sine–Gordon model==
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| A supersymmetric extension of the sine–Gordon model also exists. Integrability preserving boundary conditions for this extension can be found as well.
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| == See also ==
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| * [[Josephson effect]]
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| * [[Fluxon]]
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| * [[Shape waves]]
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| == References ==
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| {{reflist|2}}
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| *{{citation|first=E. |last=Bour|title= Théorie de la déformation des surfaces|journal= J. Ecole Imperiale Polytechnique |volume=19|pages= 1–48 |year=1862}}
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| *{{citation|last1=Frenkel|first1=J. |last2=Kontorova|first2=T. |year=1939|title=On the theory of plastic deformation and twinning|journal=Journal of Physics (USSR)|volume=1|pages=137–149}}
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| ==External links==
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| * [http://eqworld.ipmnet.ru/en/solutions/npde/npde2106.pdf Sine–Gordon Equation] at EqWorld: The World of Mathematical Equations.
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| * [http://eqworld.ipmnet.ru/en/solutions/npde/npde2105.pdf Sinh–Gordon Equation] at EqWorld: The World of Mathematical Equations.
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| * [http://www.primat.mephi.ru/wiki/ow.asp?Sine-Gordon_equation Sine–Gordon equation] at NEQwiki, the nonlinear equations encyclopedia.
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| {{DEFAULTSORT:Sine-Gordon Equation}}
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| [[Category:Solitons]]
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| [[Category:Differential geometry]]
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| [[Category:Surfaces]]
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| [[Category:Exactly solvable models]]
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