Sine-Gordon equation

2025-08-10T19:18:39Z

2607:FA49:6B64:200:8DE0:6299:271B:BEAE: Fixed lower bound for superrenormalizable to non strict inequality.

{{Short description|Nonlinear partial differential equation}}
The '''sine-Gordon equation''' is a second-order [[nonlinear partial differential equation]] for a function <math>\varphi</math> dependent on two variables typically denoted <math>x</math> and <math>t</math>, involving the [[wave operator]] and the [[sine and cosine|sine]] of <math>\varphi</math>.

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 constant [[Gaussian curvature]] −1 in [[3-dimensional space]].<ref name="Bour1862">{{cite journal
| last1 = Bour
| first1 = Edmond
| title = Theorie de la deformation des surfaces
| journal = Journal de l'École impériale polytechnique
| volume = 22
| issue = 39
| pages = 1–148
| year = 1862
| url = https://gallica.bnf.fr/ark:/12148/bpt6k433694t
| oclc = 55567842
}}</ref> The equation was rediscovered by {{harvs|txt|last1=Frenkel|last2= Kontorova|year=1939|author-link=Yakov Frenkel|first1=Yakov|first2=Tatyana}} in their study of [[crystal dislocation]]s known as the [[Frenkel–Kontorova model]].<ref name="FrenkelKontorova1939">{{cite journal
|vauthors=Frenkel J, Kontorova T | title = On the theory of plastic deformation and twinning
| journal = Izvestiya Akademii Nauk SSSR. Seriya Fizicheskaya
| volume = 1
| pages = 137–149
| year = 1939
}}</ref>

This equation attracted a lot of attention in the 1970s due to the presence of [[soliton]] solutions,<ref name="hirota">{{cite journal |last1=Hirota |first1=Ryogo |title=Exact Solution of the Sine-Gordon Equation for Multiple Collisions of Solitons |journal=Journal of the Physical Society of Japan |date=November 1972 |volume=33 |issue=5 |pages=1459–1463 |doi=10.1143/JPSJ.33.1459|bibcode=1972JPSJ...33.1459H }}</ref> and is an example of an [[integrable system|integrable PDE]]. Among well-known integrable PDEs, the sine-Gordon equation is the only ''relativistic'' system due to its [[Lorentz invariance]].

== Realizations of the sine-Gordon equation ==

=== Differential geometry ===
This is the first derivation of the equation, by Bour (1862).

There are two equivalent forms of the sine-Gordon equation. In the ([[real number|real]]) ''space-time coordinates'', denoted <math>(x,t)</math>, the equation reads:<ref name="Rajaraman1989">{{cite book
| last = Rajaraman
| first = R.
| title = Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory
| publisher = North-Holland
| series = North-Holland Personal Library
| volume = 15
| pages = 34–45
| year = 1989
| isbn = 978-0-444-87047-6
}}</ref>

: <math>\varphi_{tt} - \varphi_{xx} + \sin\varphi = 0,</math>

where partial derivatives are denoted by subscripts. Passing to the [[light-cone coordinates]] (''u'', ''v''), akin to ''asymptotic coordinates'' where

<math display="block">u = \frac{x + t}{2}, \quad v = \frac{x - t}{2},</math>

the equation takes the form<ref name="Polyanin2004">{{cite book
| last = Polyanin
| first = Andrei D.
| author2=Valentin F. Zaitsev
| title = Handbook of Nonlinear Partial Differential Equations
| publisher = Chapman & Hall/CRC Press
| pages = 470–492
| year = 2004
| isbn = 978-1-58488-355-5
}}</ref>

<math display="block">\varphi_{uv} = \sin\varphi.</math>

This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of [[differential geometry of surfaces|surfaces]] of constant [[Gaussian curvature]] ''K'' = −1, also called [[pseudospherical surface]]s.

Consider an arbitrary pseudospherical surface. Across every point on the surface there are two [[Asymptotic curve|asymptotic curves]]. This allows us to construct a distinguished coordinate system for such a surface, in which ''u'' = constant, ''v'' = constant are the asymptotic lines, and the coordinates are incremented by the [[arc length]] on the surface. At every point on the surface, let <math>\varphi</math> be the angle between the asymptotic lines.

The [[first fundamental form]] of the surface is

<math display="block">ds^2 = du^2 + 2\cos\varphi \,du\,dv + dv^2,</math>

and the [[second fundamental form]] is<math display="block">L = N = 0, M = \sin \varphi</math>and the [[Gauss–Codazzi equation]] is<math display="block">\varphi_{uv} = \sin\varphi.</math>Thus, any pseudospherical surface gives rise to a solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily [[singular curve|singular]] due to the [[Hilbert's theorem (differential geometry)|Hilbert embedding theorem]]. In the simplest case, ''the'' [[pseudosphere]], also known as the tractroid, corresponds to a static one-soliton, but the tractroid has a singular cusp at its equator.

Conversely, one can start with a solution to the sine-Gordon equation to obtain a pseudosphere uniquely up to [[rigid transformation]]s. There is a theorem, sometimes called the ''fundamental theorem of surfaces'', that if a pair of matrix-valued bilinear forms satisfy the Gauss–Codazzi equations, then they are the first and second fundamental forms of an embedded surface in 3-dimensional space. Solutions to the sine-Gordon equation can be used to construct such matrices by using the forms obtained above.

[[File:Deforming a pseudosphere to Dini's surface.gif|alt=A pseudosphere is deformed to a Dini surface through the Lie transform|center|thumb|500x500px|Lie transform applied to pseudosphere to obtain a [[Dini's surface|Dini surface]]]]

=== New solutions from old ===
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. Another transformation of pseudospherical surfaces is the [[squeeze mapping#Lie transform|Lie transform]] introduced by [[Sophus Lie]] in 1879, which corresponds to [[Lorentz boost]]s for solutions of the sine-Gordon equation.<ref name=terng>{{Cite journal |author=Terng, C. L., & Uhlenbeck, K. |year=2000 |journal=Notices of the AMS |volume=47 |issue=1 |title=Geometry of solitons |pages=17–25 |url=https://www.ams.org/journals/notices/200001/fea-terng.pdf}}</ref>

There are also some more straightforward ways to construct new solutions but which do not give new surfaces. Since the sine-Gordon equation is odd, the negative of any solution is another solution. However this does not give a new surface, as the sign-change comes down to a choice of direction for the normal to the surface. New solutions can be found by translating the solution: if <math>\varphi</math> is a solution, then so is <math>\varphi + 2n\pi</math> for <math>n</math> an integer.

=== Frenkel–Kontorova model ===
{{Main|Frenkel–Kontorova model}}

=== A mechanical model ===
[[File:Sine gordon 5.gif|thumb|A line of pendula, with a "breather pattern" oscillating in the middle. Unfortunately, the picture is drawn with gravity pointing ''up''.]]
Consider a line of pendula, hanging on a straight line, in constant gravity. Connect the bobs of the pendula together by a string in constant tension. Let the angle of the pendulum at location <math>x</math> be <math>\varphi</math>, then schematically, the dynamics of the line of pendulum follows Newton's second law:<math display="block">\underbrace{m\varphi_{tt}}_{\text{mass times acceleration}} = \underbrace{T \varphi_{xx}}_{\text{tension}} - \underbrace{mg \sin\varphi }_{\text{gravity}}</math>and this is the sine-Gordon equation, after scaling time and distance appropriately.

Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely <math>T\varphi_{xx}</math>, but more accurately <math>T\varphi_{xx} (1+\varphi_x^2)^{-3/2}</math>. However this does give an intuitive picture for the sine-gordon equation. One can produce exact mechanical realizations of the sine-gordon equation by more complex methods.<ref>{{Citation |last=Malomed |first=Boris A. |title=The sine-Gordon Model: General Background, Physical Motivations, Inverse Scattering, and Solitons |date=2014 |url=https://link.springer.com/10.1007/978-3-319-06722-3_1 |work=The sine-Gordon Model and its Applications |series=Nonlinear Systems and Complexity |volume=10 |pages=1–30 |editor-last=Cuevas-Maraver |editor-first=Jesús |access-date=2023-11-17 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-06722-3_1 |isbn=978-3-319-06721-6 |editor2-last=Kevrekidis |editor2-first=Panayotis G. |editor3-last=Williams |editor3-first=Floyd|url-access=subscription }}</ref>

== Naming ==

The name "sine-Gordon equation" is a pun on the well-known [[Klein–Gordon equation]] in physics:<ref name="Rajaraman1989"/>

<math display="block">\varphi_{tt} - \varphi_{xx} + \varphi = 0.</math>

The sine-Gordon equation is the [[Euler–Lagrange equation]] of the field whose [[Lagrangian density]] is given by

<math display="block">\mathcal{L}_\text{SG}(\varphi) = \frac{1}{2} (\varphi_t^2 - \varphi_x^2) - 1 + \cos\varphi.</math>

Using the [[Taylor series]] expansion of the [[cosine]] in the Lagrangian,

<math display="block">\cos(\varphi) = \sum_{n=0}^\infty \frac{(-\varphi^2)^n}{(2n)!},</math>

it can be rewritten as the [[Scalar field theory#Linear .28free.29 theory|Klein–Gordon Lagrangian]] plus higher-order terms:

<math display="block">
\begin{align}
\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)!} \\
&= \mathcal{L}_\text{KG}(\varphi) + \sum_{n=2}^\infty \frac{(-\varphi^2)^n}{(2n)!}.
\end{align}
</math>

== Soliton solutions ==

An interesting feature of the sine-Gordon equation is the existence of [[soliton]] and multisoliton solutions.

=== 1-soliton solutions ===
The sine-Gordon equation has the following 1-[[soliton]] solutions:

<math display="block">\varphi_\text{soliton}(x, t) := 4 \arctan \left(e^{m \gamma (x - vt) + \delta}\right),</math>

where

<math display="block">\gamma^2 = \frac{1}{1 - v^2},</math>

and the slightly more general form of the equation is assumed:

<math display="block">\varphi_{tt} - \varphi_{xx} + m^2 \sin\varphi = 0.</math>

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 constant solution <math>\varphi = 0</math> to an adjacent constant solution <math>\varphi = 2\pi</math>. The states <math>\varphi \cong 2\pi n</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 (vacuum) solution and the integration of the resulting first-order differentials:

<math display="block">\varphi'_u = \varphi_u + 2\beta \sin\frac{\varphi' + \varphi}{2},</math>

<math display="block">\varphi'_v = -\varphi_v + \frac{2}{\beta} \sin\frac{\varphi' - \varphi}{2} \text{ with } \varphi = \varphi_0 = 0</math>

for all time.

The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model introduced by Julio Rubinstein in 1970.<ref name="Rubinstein1970">{{cite journal
| last1 = Rubinstein
| first1 = Julio
| title = Sine-Gordon equation
| journal = Journal of Mathematical Physics
| volume = 11
| issue = 1
| pages = 258–266
| year = 1970
| doi = 10.1063/1.1665057
| bibcode = 1970JMP....11..258R
}}</ref> Here we take a clockwise ([[Right-hand rule|left-handed]]) twist of the elastic ribbon to be a kink with topological charge <math>\theta_\text{K} = -1</math>. The alternative counterclockwise ([[Right-hand rule|right-handed]]) twist with topological charge <math>\theta_\text{AK} = +1</math> will be an antikink.

{|
|-
| [[Image:Sine gordon 1.gif|frame|Traveling ''kink'' soliton represents a propagating clockwise twist.<ref name="Georgiev2004">{{cite journal
|author=Georgiev D. D. |author2=Papaioanou S. N. |author3=Glazebrook J. F.
| title = Neuronic system inside neurons: molecular biology and biophysics of neuronal microtubules
| journal = Biomedical Reviews
| volume = 15
| pages = 67–75
| year = 2004
| url = http://cogprints.org/4364/
| doi = 10.14748/bmr.v15.103
| doi-access = free
}}</ref>]]
| [[Image:Sine gordon 2.gif|frame|Traveling ''antikink'' soliton represents a propagating counterclockwise twist.<ref name="Georgiev2004"/>]]
|}

[[File:Static one-soliton.png|thumb|Static 1-soliton solution <math>4\arctan e^x</math>|none|474x474px]]

=== 2-soliton solutions ===

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
| last = Rogers | first = C.
| author2 = W. K. Schief
| title = Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
| publisher = [[Cambridge University Press]]
| location = New York
| series = Cambridge Texts in Applied Mathematics
| year = 2002
| isbn = 978-0-521-01288-1
}}</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 an interaction is called an [[elastic collision]].

The kink-kink solution is given by
<math display=block>\varphi_{K/K}(x,t) = 4 \arctan \left(\frac{v \sinh \frac{x}{\sqrt{1 - v^2}}}{\cosh \frac{vt}{\sqrt{1 - v^2}}}\right)</math>

while the kink-antikink solution is given by
<math display=block>\varphi_{K/AK}(x,t) = 4 \arctan \left(\frac{v \cosh \frac{x}{\sqrt{1 - v^2}}}{\sinh \frac{vt}{\sqrt{1 - v^2}}}\right)</math>

{|
|-
| [[Image:Sine gordon 3.gif|frame|''Antikink-kink'' collision.<ref name="Georgiev2004"/>]]
| [[Image:Sine gordon 4.gif|frame|''Kink-kink'' collision.<ref name="Georgiev2004"/>]]
|}

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. E., Vasiliev A. A., Dmitriev S. V. ''[http://homepages.tversu.ru/~s000154/collision/main.html Solitons and Soliton Collisions]''.</ref>

The standing breather solution is given by
<math display=block>\varphi(x,t) = 4 \arctan\left(\frac{\sqrt{1-\omega^2}\;\cos(\omega t)}{\omega\;\cosh(\sqrt{1-\omega^2}\; x)}\right).</math>

{|
|-
| [[Image:Sine gordon 5.gif|frame|The ''standing breather'' is an oscillating coupled kink-antikink soliton.<ref name="Georgiev2004"/>]]
| [[Image:Sine gordon 6.gif|frame|''Large-amplitude moving breather''.<ref name="Georgiev2004"/>]]
|}

{|
|-
| [[Image:Sine gordon 7.gif|frame|''Small-amplitude moving breather''{{snd}} looks exotic, but essentially has a breather envelope.<ref name="Georgiev2004"/>]]
|}

=== 3-soliton solutions ===

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,
the shift of the breather <math>\Delta_\text{B}</math> is given by

<math display="block">\Delta_\text{B} =\frac{2\operatorname{artanh}\sqrt{(1 - \omega^2)(1 - v_\text{K}^2)}}{\sqrt{1 - \omega^2}},</math>

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>.

{|
|-
| [[Image:Sine gordon 8.gif|frame|Collision of ''moving kink'' and ''standing breather''.<ref name="Georgiev2004"/>]]
| [[Image:Sine gordon 9.gif|frame|Collision of ''moving antikink'' and ''standing breather''.<ref name="Georgiev2004"/>]]
|}

==Bäcklund transformation==
{{See also|Bäcklund transform}}
Suppose that <math>\varphi</math> is a solution of the sine-Gordon equation
<math display="block"> \varphi_{uv} = \sin \varphi.\,</math>

Then the system
<math display="block">\begin{align}
\psi_u & = \varphi_u + 2a \sin \Bigl( \frac{\psi+\varphi}{2} \Bigr) \\
\psi_v & = -\varphi_v + \frac{2}{a} \sin \Bigl( \frac{\psi-\varphi}{2} \Bigr)
\end{align} \,\!</math>
where ''a'' is an arbitrary parameter, is solvable for a function <math>\psi</math> which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform, as both <math>\varphi</math> and <math>\psi</math> are solutions to the same equation, that is, the sine-Gordon equation.

By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.

For example, if <math>\varphi</math> is the trivial solution <math>\varphi \equiv 0</math>, then <math>\psi</math> is the one-soliton solution with <math>a</math> related to the boost applied to the soliton.

==Topological charge and energy==
The '''topological charge''' or '''winding number''' of a solution <math>\varphi</math> is
<math display=block>N = \frac{1}{2\pi} \int_\mathbb{R} d\varphi = \frac{1}{2\pi} \left[\varphi(x = \infty, t) - \varphi(x = -\infty, t)\right].</math>
The '''energy''' of a solution <math>\varphi</math> is
<math display="block">E = \int_\mathbb{R}dx \left(\frac{1}{2}( \varphi_t^2 + \varphi_x^2) + m^2(1 - \cos\varphi)\right)</math>where a constant energy density has been added so that the potential is non-negative. With it the first two terms in the Taylor expansion of the potential coincide with the potential of a massive scalar field, as mentioned in the naming section; the higher order terms can be thought of as interactions.

The topological charge is conserved if the energy is finite. The topological charge does not determine the solution, even up to Lorentz boosts. Both the trivial solution and the soliton-antisoliton pair solution have <math>N = 0</math>.

==Zero-curvature formulation==
The sine-Gordon equation is equivalent to the [[curvature form|curvature]] of a particular <math>\mathfrak{su}(2)</math>-[[principal connection|connection]] on <math>\mathbb{R}^2</math> being equal to zero.<ref name="SIT">{{cite book |last1=Dunajski |first1=Maciej |title=Solitons, instantons, and twistors |date=2010 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-857063-9 |page=49}}</ref>

Explicitly, with coordinates <math>(u,v)</math> on <math>\mathbb{R}^2</math>, the connection components <math>A_\mu</math> are given by
<math display=block>A_u = \begin{pmatrix}i\lambda & \frac{i}{2}\varphi_u \\ \frac{i}{2}\varphi_u & -i\lambda\end{pmatrix} = \frac{1}{2}\varphi_u i\sigma_1 + \lambda i\sigma_3,</math>
<math display=block>A_v = \begin{pmatrix}-\frac{i}{4\lambda}\cos\varphi & -\frac{1}{4\lambda}\sin\varphi \\ \frac{1}{4\lambda}\sin\varphi & \frac{i}{4\lambda}\cos\varphi\end{pmatrix} = -\frac{1}{4\lambda}i\sin\varphi\sigma_2 - \frac{1}{4\lambda}i\cos\varphi\sigma_3,</math>
where the <math>\sigma_i</math> are the [[Pauli matrices]].
Then the zero-curvature equation
<math display=block>\partial_v A_u - \partial_u A_v + [A_u, A_v] = 0</math>

is equivalent to the sine-Gordon equation <math>\varphi_{uv} = \sin\varphi</math>. The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined <math>F_{\mu\nu} = [\partial_\mu - A_\mu, \partial_\nu - A_\nu]</math>.

The pair of matrices <math>A_u</math> and <math>A_v</math> are also known as a [[Lax pair]] for the sine-Gordon equation, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.

==Related equations==

The '''{{visible anchor|sinh-Gordon equation}}''' is given by<ref>{{cite book |first1=Andrei D. |last1=Polyanin |first2=Valentin F. |last2=Zaitsev |title=Handbook of Nonlinear Partial Differential Equations |date=16 December 2011 |location=Boca Raton |edition=Second |page=485 |publisher=CRC Press |isbn=978-1-4200-8723-9 }}</ref>

<math display="block">\varphi_{xx} - \varphi_{tt} = \sinh\varphi.</math>

This is the [[Euler–Lagrange equation]] of the [[Lagrangian (field theory)|Lagrangian]]

<math display="block">\mathcal{L} = \frac{1}{2} (\varphi_t^2 - \varphi_x^2) - \cosh\varphi.</math>

Another closely related equation is the '''elliptic sine-Gordon equation''' or '''Euclidean sine-Gordon equation''', given by

<math display="block">\varphi_{xx} + \varphi_{yy} = \sin\varphi,</math>

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''.

The '''elliptic sinh-Gordon equation''' may be defined in a similar way.

Another similar equation comes from the Euler–Lagrange equation for [[Liouville field theory]]

<math display=block>\varphi_{xx} - \varphi_{tt} = 2e^{2\varphi}.</math>

A generalization is given by [[Toda field theory]].<ref name=Yuanxi>{{cite journal |last=Yuanxi |first=Xie |author2=Tang, Jiashi |title=A unified method for solving sinh-Gordon–type equations |journal=Il Nuovo Cimento B |date=February 2006 |volume=121 |issue=2 |pages=115–121 |doi=10.1393/ncb/i2005-10164-6 |bibcode=2006NCimB.121..115X }}</ref> More precisely, Liouville field theory is the Toda field theory for the finite [[Kac–Moody algebra]] <math>\mathfrak{sl}_2</math>, while sin(h)-Gordon is the Toda field theory for the [[affine Kac–Moody algebra]] <math>\hat \mathfrak{sl}_2</math>.

==Infinite volume and on a half line==

One can also consider the sine-Gordon model on a circle,<ref name="McKean1981">{{cite journal
| last1 = McKean
| first1 = H. P.
| title = The sine-Gordon and sinh-Gordon equations on the circle
| journal = Communications on Pure and Applied Mathematics
| volume = 34
| issue = 2
| pages = 197–257
| year = 1981
| doi = 10.1002/cpa.3160340204
}}</ref> on a line segment, or on a half line.<ref name="Bowcock2007">{{cite journal
| last1 = Bowcock
| first1 = Peter
| last2 = Tzamtzis
| first2 = Georgios
| title = The complex sine-Gordon model on a half line
| journal = Journal of High Energy Physics
| volume = 2007
| issue = 3
| page = 047
| year = 2007
| doi = 10.1088/1126-6708/2007/03/047
| arxiv = hep-th/0203139
| bibcode = 2007JHEP...03..047B
| s2cid = 119501952
}}</ref> It is possible to find boundary conditions which preserve the integrability of the model.<ref name="Bowcock2007"/> On a half line the spectrum contains ''boundary bound states'' in addition to the solitons and breathers.<ref name="Bowcock2007"/>

==Quantum sine-Gordon model==

In [[quantum field theory]] the sine-Gordon model contains a parameter that can be identified with the [[Planck constant]]. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of [[breather]]s.<ref name="Korepin1979">{{cite journal
| last1 = Korepin
| first1 = V. E.
| title = Direct calculation of the S matrix in the massive Thirring model
| journal = Theoretical and Mathematical Physics
| volume = 41
| issue = 2
| pages = 953–967
| year = 1979
| doi = 10.1007/bf01028501
| bibcode = 1979TMP....41..953K
| s2cid = 121527379
}}</ref><ref name="Takada1981">{{cite journal
| last1 = Takada
| first1 = Satoshi
| last2 = Misawa
| first2 = Susumu
| title = The Quantum Sine-Gordon Model and the Fermi-Bose Relation
| journal = Progress of Theoretical Physics
| volume = 66
| issue = 1
| pages = 101–117
| year = 1981
| doi = 10.1143/ptp.66.101
| bibcode = 1981PThPh..66..101T
}}</ref><ref name="Bogoliubov1985">{{cite journal
| last1 = Bogoliubov
| first1 = N. M.
| last2 = Korepin
| first2 = V. E.
| last3 = Izergin
| first3 = A. G.
| title = Structure of the vacuum in the quantum sine-Gordon model
| journal = Physics Letters B
| volume = 159
| issue = 4
| pages = 345–347
| year = 1985
| doi = 10.1016/0370-2693(85)90264-3
| bibcode = 1985PhLB..159..345B
}}</ref> The number of breathers depends on the value of the parameter. Multiparticle production cancels on mass shell.

Semi-classical quantization of the sine-Gordon model was done by [[Ludwig Faddeev]] and [[Vladimir Korepin]].<ref name="Faddeev1978">{{cite journal
| last1 = Faddeev
| first1 = L. D.
| last2 = Korepin
| first2 = V. E.
| title = Quantum theory of solitons
| journal = Physics Reports
| volume = 42
| issue = 1
| pages = 1–87
| year = 1978
| doi = 10.1016/0370-1573(78)90058-3
| bibcode = 1978PhR....42....1F
}}</ref> The exact quantum [[scattering matrix]] was discovered by [[Alexander Zamolodchikov]].<ref name="Zamolodchikov1978">{{cite journal
| last1 = Zamolodchikov
| first1 = Alexander B.
| last2 = Zamolodchikov
| first2 = Alexey B.
| title = Relativistic factorized S-matrix in two dimensions having O(N) isotopic symmetry
| journal = Nuclear Physics B
| volume = 133
| issue = 3
| pages = 525–535
| year = 1978
| doi = 10.1016/0550-3213(78)90239-0
| bibcode = 1978NuPhB.133..525Z
}}</ref>
This model is [[S-duality|S-dual]] to the [[Thirring model]], as discovered by [[Sidney Coleman|Coleman]].<ref name="coleman">{{cite journal |last1=Coleman |first1=Sidney |title=Quantum sine-Gordon equation as the massive Thirring model |journal=Physical Review D |date=15 April 1975 |volume=11 |issue=8 |pages=2088–2097 |doi=10.1103/PhysRevD.11.2088 |bibcode=1975PhRvD..11.2088C |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.11.2088 |access-date=27 January 2023|url-access=subscription }}</ref> This is sometimes known as the Coleman correspondence and serves as an example of boson-fermion correspondence in the interacting case. This article also showed that the constants appearing in the model behave nicely under [[renormalization]]: there are three parameters <math>\alpha_0, \beta</math> and <math>\gamma_0</math>. Coleman showed <math>\alpha_0</math> receives only a multiplicative correction, <math>\gamma_0</math> receives only an additive correction, and <math>\beta</math> is not renormalized. Further, for a critical, non-zero value <math>\beta = \sqrt{4\pi}</math>, the theory is in fact dual to a ''free'' massive [[Dirac equation#Lagrangian formulation|Dirac field theory]].

The quantum sine-Gordon equation should be modified so the exponentials become [[vertex operator]]s

<math display="block">\mathcal{L}_{QsG} = \frac{1}{2} \partial_\mu \varphi \partial^\mu \varphi + \frac{1}{2}m_0^2\varphi^2 - \alpha(V_\beta + V_{-\beta})</math>

with <math>V_\beta = :e^{i\beta\varphi}:</math>, where the semi-colons denote [[normal ordering]]. A possible mass term is included.

=== Regimes of renormalizability ===
For different values of the parameter <math>\beta^2</math>, the [[renormalization|renormalizability]] properties of the sine-Gordon theory change.<ref>{{cite journal |last1=Fröb |first1=Markus B. |last2=Cadamuro |first2=Daniela |title=Local Operators in the Sine-Gordon Model: <math>\partial _\mu \phi \, \partial _\nu \phi </math> and the Stress Tensor |journal=Annales Henri Poincaré |date=2025 |doi=10.1007/s00023-025-01565-z |arxiv=2205.09223 }}</ref> The identification of these regimes is attributed to [[Jürg Fröhlich]].

The '''finite regime''' is <math>\beta^2 < 4\pi</math>, where no [[counterterm]]s are needed to render the theory well-posed. The '''super-renormalizable regime''' is <math>4\pi \leq \beta^2 < 8\pi</math>, where a finite number of counterterms are needed to render the theory well-posed. More counterterms are needed for each threshold <math>\frac{n}{n+1}8\pi</math> passed.<ref>{{cite arXiv |last1=Chandra |first1=Ajay |last2=Hairer |first2=Martin |last3=Shen |first3=Hao |title=The dynamical sine-Gordon model in the full subcritical regime |date=2018 |class=math.PR |eprint=1808.02594 }}</ref> For <math>\beta^2 > 8\pi</math>, the theory becomes ill-defined {{harvs|last=Coleman|year=1975}}. The boundary values are <math>\beta^2 = 4\pi</math> and <math>\beta^2 = 8\pi</math>, which are respectively the free fermion point, as the theory is dual to a free fermion via the Coleman correspondence, and the self-dual point, where the vertex operators form an [[affine Kac–Moody algebra|affine sl<sub>2</sub> subalgebra]], and the theory becomes strictly renormalizable (renormalizable, but not super-renormalizable).

==Stochastic sine-Gordon model==
The '''stochastic''' or '''dynamical sine-Gordon model''' has been studied by [[Martin Hairer]] and Hao Shen
<ref name="hairer-shen">
{{cite journal
|last1=Hairer
|first1=Martin
|last2=Shen
|first2=Hao
|title=The Dynamical Sine-Gordon Model
|journal=Communications in Mathematical Physics
|date=February 2016
|volume=341
|issue=3
|pages=933–989
|doi=10.1007/s00220-015-2525-3
|arxiv=1409.5724
|bibcode=2016CMaPh.341..933H
|s2cid=253750515
|url=https://link.springer.com/article/10.1007/s00220-015-2525-3
|access-date=14 May 2023}}</ref>
allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.

The equation is
<math display = block>\partial_t u = \frac{1}{2}\Delta u + c\sin(\beta u + \theta) + \xi,</math>
where <math>c, \beta, \theta</math> are real-valued constants, and <math>\xi</math> is space-time [[white noise]]. The space dimension is fixed to 2. In the proof of existence of solutions, the thresholds <math>\beta^2 = \frac{n}{n+1}8\pi</math> again play a role in determining convergence of certain terms.

==Supersymmetric sine-Gordon model==

A supersymmetric extension of the sine-Gordon model also exists.<ref name="Inami1995">{{cite journal
| last1 = Inami
| first1 = Takeo
| last2 = Odake
| first2 = Satoru
| last3 = Zhang
| first3 = Yao-Zhong
| title = Supersymmetric extension of the sine-Gordon theory with integrable boundary interactions
| journal = Physics Letters B
| volume = 359
| issue = 1
| pages = 118–124
| year = 1995
| doi = 10.1016/0370-2693(95)01072-X
| arxiv = hep-th/9506157
| bibcode = 1995PhLB..359..118I
| s2cid = 18230581
}}</ref> Integrability preserving boundary conditions for this extension can be found as well.<ref name="Inami1995"/>

==Physical applications==
The sine-Gordon model arises as the continuum limit of the [[Frenkel–Kontorova model]] which models crystal dislocations.

Dynamics in [[long Josephson junction]]s are well-described by the sine-Gordon equations, and conversely provide a useful experimental system for studying the sine-Gordon model.<ref name="MU">{{cite book
|last1=Mazo |first1=Juan J.
|last2=Ustinov |first2=Alexey V.
|title=The sine-Gordon Model and its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics |date=2014
|publisher=Springer International Publishing |isbn=978-3-319-06722-3
|pages=155–175
|chapter-url=https://link.springer.com/chapter/10.1007/978-3-319-06722-3_7
|access-date=22 August 2023 |language=en
|chapter=The sine-Gordon Equation in Josephson-Junction Arrays|doi=10.1007/978-3-319-06722-3_7
}}</ref>

The sine-Gordon model is in the same [[universality class]] as the [[effective action]] for a [[Coulomb gas]] of [[quantum vortex|vortices]] and anti-vortices in the continuous [[classical XY model]], which is a model of magnetism.<ref>{{cite journal
|last1=José |first1=Jorge
|title=Sine-Gordon theory and the classical two-dimensional x − y model
|journal=Physical Review D |date=15 November 1976
|volume=14 |issue=10
|pages=2826–2829
|doi=10.1103/PhysRevD.14.2826
|bibcode=1976PhRvD..14.2826J }}</ref><ref>{{cite journal
|last1=Fröhlich |first1=Jürg
|title=Classical and quantum statistical mechanics in one and two dimensions: Two-component Yukawa — and Coulomb systems
|journal=Communications in Mathematical Physics
|date=October 1976
|volume=47 |issue=3 |pages=233–268
|doi=10.1007/BF01609843
|bibcode=1976CMaPh..47..233F
|s2cid=120798940
|url=http://projecteuclid.org/euclid.cmp/1103899760 }}</ref> The [[Kosterlitz–Thouless transition]] for vortices can therefore be derived from a [[renormalization group]] analysis of the sine-Gordon field theory.<ref>{{cite journal
|last1=Ohta |first1=T.
|last2=Kawasaki |first2=K.
|title=Renormalization Group Theory of the Interfacial Roughening Transition
|journal=Progress of Theoretical Physics
|date=1 August 1978
|volume=60 |issue=2
|pages=365–379
|doi=10.1143/PTP.60.365|bibcode=1978PThPh..60..365O |doi-access=free
}}</ref><ref>{{cite journal
|last1=Kogut |first1=John B.
|title=An introduction to lattice gauge theory and spin systems
|journal=Reviews of Modern Physics
|date=1 October 1979 |volume=51 |issue=4 |pages=659–713
|doi=10.1103/RevModPhys.51.659|bibcode=1979RvMP...51..659K }}</ref>

The sine-Gordon equation also arises as the formal continuum limit of a different model of magnetism, the [[quantum Heisenberg model]], in particular the XXZ model.<ref>{{cite arXiv |last1=Faddeev |first1=L. D.
|title=How Algebraic Bethe Ansatz works for integrable model |date=1996
|eprint=hep-th/9605187 }}</ref>

== See also ==
* [[Josephson effect]]
* [[Fluxon]]
* [[Shape waves]]

== References ==
{{reflist|2}}

== External links ==
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde2106.pdf sine-Gordon equation] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde2105.pdf Sinh-Gordon Equation] at EqWorld: The World of Mathematical Equations.
* [http://www.primat.mephi.ru/wiki/ow.asp?Sine-Gordon_equation sine-Gordon equation] {{Webarchive|url=https://web.archive.org/web/20120316101044/http://www.primat.mephi.ru/wiki/ow.asp?Sine%2DGordon%5Fequation |date=2012-03-16 }} at NEQwiki, the nonlinear equations encyclopedia.

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Sine-Gordon equation