The sine-Gordon equation is a nonlinear hyperbolic partial differential equation for a function ${\displaystyle \varphi }$ dependent on two variables typically denoted ${\displaystyle x}$ and ${\displaystyle t}$, involving the wave operator and the sine of ${\displaystyle \varphi }$.

It was originally introduced by Edmond Bour (1862) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space.^[1] The equation was rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model.^[2]

This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions,^[3] and is an example of an 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) space-time coordinates, denoted ${\displaystyle (x,t)}$, the equation reads:^[4]

${\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}$

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

${\displaystyle u={\frac {x+t}{2)),\quad v={\frac {x-t}{2)),}$

the equation takes the form^[5]

${\displaystyle \varphi _{uv}=\sin \varphi .}$

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

Consider an arbitrary pseudospherical surface. Across every point on the surface there are two 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 ${\displaystyle \varphi }$ be the angle between the asymptotic lines.

The first fundamental form of the surface is

${\displaystyle ds^{2}=du^{2}+2\cos \varphi \,du\,dv+dv^{2},}$

and the second fundamental form is

$${\displaystyle L=N=0,M=\sin \varphi }$$

$${\displaystyle \varphi _{uv}=\sin \varphi .}$$

Conversely, one can start with a solution to the sine-Gordon equation to obtain a pseudosphere uniquely up to rigid transformations. 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.

New solutions from old

The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Bianchi and Bäcklund led to the discovery of Bäcklund transformations. Another transformation of pseudospherical surfaces is the Lie transform introduced by Sophus Lie in 1879, which corresponds to Lorentz boosts for solutions of the sine-Gordon equation.^[6]

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 ${\displaystyle \varphi }$ is a solution, then so is ${\displaystyle \varphi +2n\pi }$ for ${\displaystyle n}$ an integer.

Frenkel–Kontorova model

A mechanical model

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 ${\displaystyle x}$ be ${\displaystyle \varphi }$, then schematically, the dynamics of the line of pendulum follows Newton's second law:

$${\displaystyle \underbrace {m\varphi _{tt)) _{\text{mass times acceleration))=\underbrace {T\varphi _{xx)) _{\text{tension))-\underbrace {mg\sin \varphi } _{\text{gravity))}$$

Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely ${\displaystyle T\varphi _{xx))$, but more accurately ${\displaystyle T\varphi _{xx}(1+\varphi _{x}^{2})^{-3/2))$. 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.^[7]

Naming

The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics:^[4]

${\displaystyle \varphi _{tt}-\varphi _{xx}+\varphi =0.}$

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

${\displaystyle {\mathcal {L))_{\text{SG))(\varphi )={\frac {1}{2))(\varphi _{t}^{2}-\varphi _{x}^{2})-1+\cos \varphi .}$

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

${\displaystyle \cos(\varphi )=\sum _{n=0}^{\infty }{\frac {(-\varphi ^{2})^{n)){(2n)!)),}$

it can be rewritten as the Klein–Gordon Lagrangian plus higher-order terms:

${\displaystyle {\begin{aligned}{\mathcal {L))_{\text{SG))(\varphi )&={\frac {1}{2))(\varphi _{t}^{2}-\varphi _{x}^{2})-{\frac {\varphi ^{2)){2))+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n)){(2n)!))\\&={\mathcal {L))_{\text{KG))(\varphi )+\sum _{n=2}^{\infty }{\frac {(-\varphi ^{2})^{n)){(2n)!)).\end{aligned))}$

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:

${\displaystyle \varphi _{\text{soliton))(x,t):=4\arctan \left(e^{m\gamma (x-vt)+\delta }\right),}$

where

${\displaystyle \gamma ^{2}={\frac {1}{1-v^{2))},}$

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

${\displaystyle \varphi _{tt}-\varphi _{xx}+m^{2}\sin \varphi =0.}$

The 1-soliton solution for which we have chosen the positive root for ${\displaystyle \gamma }$ is called a kink and represents a twist in the variable ${\displaystyle \varphi }$ which takes the system from one constant solution ${\displaystyle \varphi =0}$ to an adjacent constant solution ${\displaystyle \varphi =2\pi }$. The states ${\displaystyle \varphi \cong 2\pi n}$ 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 ${\displaystyle \gamma }$ 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:

${\displaystyle \varphi '_{u}=\varphi _{u}+2\beta \sin {\frac {\varphi '+\varphi }{2)),}$

${\displaystyle \varphi '_{v}=-\varphi _{v}+{\frac {2}{\beta ))\sin {\frac {\varphi '-\varphi }{2)){\text{ with ))\varphi =\varphi _{0}=0}$

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.^[8] Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge ${\displaystyle \theta _{\text{K))=-1}$. The alternative counterclockwise (right-handed) twist with topological charge ${\displaystyle \theta _{\text{AK))=+1}$ will be an antikink.

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.^[10] 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 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

$${\displaystyle \varphi _{K/K}(x,t)=4\arctan \left({\frac {v\sinh {\frac {x}{\sqrt {1-v^{2))))}{\cosh {\frac {vt}{\sqrt {1-v^{2))))))\right)}$$

while the kink-antikink solution is given by

$${\displaystyle \varphi _{K/AK}(x,t)=4\arctan \left({\frac {v\cosh {\frac {x}{\sqrt {1-v^{2))))}{\sinh {\frac {vt}{\sqrt {1-v^{2))))))\right)}$$

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.^[11]

The standing breather solution is given by

$${\displaystyle \varphi (x,t)=4\arctan \left({\frac ((\sqrt {1-\omega ^{2))}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2))}\;x)))\right).}$$

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 ${\displaystyle \Delta _{\text{B))}$ is given by

${\displaystyle \Delta _{\text{B))={\frac {2\operatorname {artanh} {\sqrt {(1-\omega ^{2})(1-v_{\text{K))^{2})))}{\sqrt {1-\omega ^{2)))),}$

where ${\displaystyle v_{\text{K))}$ is the velocity of the kink, and ${\displaystyle \omega }$ is the breather's frequency.^[11] If the old position of the standing breather is ${\displaystyle x_{0))$, after the collision the new position will be ${\displaystyle x_{0}+\Delta _{\text{B))}$.

Bäcklund transformation

Suppose that ${\displaystyle \varphi }$ is a solution of the sine-Gordon equation

${\displaystyle \varphi _{uv}=\sin \varphi .\,}$

Then the system

${\displaystyle {\begin{aligned}\psi _{u}&=\varphi _{u}+2a\sin {\Bigl (}{\frac {\psi +\varphi }{2)){\Bigr )}\\\psi _{v}&=-\varphi _{v}+{\frac {2}{a))\sin {\Bigl (}{\frac {\psi -\varphi }{2)){\Bigr )}\end{aligned))\,\!}$

where a is an arbitrary parameter, is solvable for a function ${\displaystyle \psi }$ which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform, as both ${\displaystyle \varphi }$ and ${\displaystyle \psi }$ 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 ${\displaystyle \varphi }$ is the trivial solution ${\displaystyle \varphi \equiv 0}$, then ${\displaystyle \psi }$ is the one-soliton solution with ${\displaystyle a}$ related to the boost applied to the soliton.

Topological charge and energy

The topological charge or winding number of a solution ${\displaystyle \varphi }$ is

$${\displaystyle N={\frac {1}{2\pi ))\int _{\mathbb {R} }d\varphi ={\frac {1}{2\pi ))\left[\varphi (x=\infty ,t)-\varphi (x=-\infty ,t)\right].}$$

${\displaystyle \varphi }$

$${\displaystyle E=\int _{\mathbb {R} }dx\left({\frac {1}{2))(\varphi _{t}^{2}+\varphi _{x}^{2})+m^{2}(1-\cos \varphi )\right)}$$

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 ${\displaystyle N=0}$.

Zero-curvature formulation

The sine-Gordon equation is equivalent to the curvature of a particular ${\displaystyle {\mathfrak {su))(2)}$-connection on ${\displaystyle \mathbb {R} ^{2))$ being equal to zero.^[12]

Explicitly, with coordinates ${\displaystyle (u,v)}$ on ${\displaystyle \mathbb {R} ^{2))$, the connection components ${\displaystyle A_{\mu ))$ are given by

$${\displaystyle A_{u}={\begin{pmatrix}i\lambda &{\frac {i}{2))\varphi _{u}\\{\frac {i}{2))\varphi _{u}&-i\lambda \end{pmatrix))={\frac {1}{2))\varphi _{u}i\sigma _{1}+\lambda i\sigma _{3},}$$

$${\displaystyle A_{v}={\begin{pmatrix}-{\frac {i}{4\lambda ))\cos \varphi &-{\frac {1}{4\lambda ))\sin \varphi \\{\frac {1}{4\lambda ))\sin \varphi &{\frac {i}{4\lambda ))\cos \varphi \end{pmatrix))=-{\frac {1}{4\lambda ))i\sin \varphi \sigma _{2}-{\frac {1}{4\lambda ))i\cos \varphi \sigma _{3},}$$

${\displaystyle \sigma _{i))$

$${\displaystyle \partial _{v}A_{u}-\partial _{u}A_{v}+[A_{u},A_{v}]=0}$$

is equivalent to the sine-Gordon equation ${\displaystyle \varphi _{uv}=\sin \varphi }$. The zero-curvature equation is so named as it corresponds to the curvature being equal to zero if it is defined ${\displaystyle F_{\mu \nu }=[\partial _{\mu }-A_{\mu },\partial _{\nu }-A_{\nu }]}$.

The pair of matrices ${\displaystyle A_{u))$ and ${\displaystyle A_{v))$ 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 sinh-Gordon equation is given by^[13]

${\displaystyle \varphi _{xx}-\varphi _{tt}=\sinh \varphi .}$

This is the Euler–Lagrange equation of the Lagrangian

${\displaystyle {\mathcal {L))={\frac {1}{2))(\varphi _{t}^{2}-\varphi _{x}^{2})-\cosh \varphi .}$

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

${\displaystyle \varphi _{xx}+\varphi _{yy}=\sin \varphi ,}$

where ${\displaystyle \varphi }$ 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 = it.

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

$${\displaystyle \varphi _{xx}-\varphi _{tt}=2e^{2\varphi }.}$$

A generalization is given by Toda field theory.^[14] More precisely, Liouville field theory is the Toda field theory for the finite Kac–Moody algebra ${\displaystyle {\mathfrak {sl))_{2))$, while sin(h)-Gordon is the Toda field theory for the affine Kac–Moody algebra ${\displaystyle {\hat {\mathfrak {sl))}_{2))$.

Infinite volume and on a half line

One can also consider the sine-Gordon model on a circle,^[15] on a line segment, or on a half line.^[16] It is possible to find boundary conditions which preserve the integrability of the model.^[16] On a half line the spectrum contains boundary bound states in addition to the solitons and breathers.^[16]

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 breathers.^[17][18][19] The number of the 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.^[20] The exact quantum scattering matrix was discovered by Alexander Zamolodchikov.^[21] This model is S-dual to the Thirring model, as discovered by Coleman. ^[22] 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 ${\displaystyle \alpha _{0},\beta }$ and ${\displaystyle \gamma _{0))$. Coleman showed ${\displaystyle \alpha _{0))$ receives only a multiplicative correction, ${\displaystyle \gamma _{0))$ receives only an additive correction, and ${\displaystyle \beta }$ is not renormalized. Further, for a critical, non-zero value ${\displaystyle \beta ={\sqrt {4\pi ))}$, the theory is in fact dual to a free massive Dirac field theory.

The quantum sine-Gordon equation should be modified so the exponentials become vertex operators

${\displaystyle {\mathcal {L))_{QsG}={\frac {1}{2))\partial _{\mu }\varphi \partial ^{\mu }\varphi +{\frac {1}{2))m_{0}^{2}\varphi ^{2}-\alpha (V_{\beta }+V_{-\beta })}$

with ${\displaystyle V_{\beta }=:e^{i\beta \varphi }:}$, where the semi-colons denote normal ordering. A possible mass term is included.

Regimes of renormalizability

For different values of the parameter ${\displaystyle \beta ^{2))$, the renormalizability properties of the sine-Gordon theory change.^[23] The identification of these regimes is attributed to Jürg Fröhlich.

The finite regime is ${\displaystyle \beta ^{2}<4\pi }$, where no counterterms are needed to render the theory well-posed. The super-renormalizable regime is ${\displaystyle 4\pi <\beta ^{2}<8\pi }$, where a finite number of counterterms are needed to render the theory well-posed. More counterterms are needed for each threshold ${\displaystyle {\frac {n}{n+1))8\pi }$ passed.^[24] For ${\displaystyle \beta ^{2}>8\pi }$, the theory becomes ill-defined (Coleman 1975). The boundary values are ${\displaystyle \beta ^{2}=4\pi }$ and ${\displaystyle \beta ^{2}=8\pi }$, 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 sl₂ 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 ^[25] allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.

The equation is

$${\displaystyle \partial _{t}u={\frac {1}{2))\Delta u+c\sin(\beta u+\theta )+\xi ,}$$

${\displaystyle c,\beta ,\theta }$

${\displaystyle \xi }$

${\displaystyle \beta ^{2}={\frac {n}{n+1))8\pi }$

Supersymmetric sine-Gordon model

A supersymmetric extension of the sine-Gordon model also exists.^[26] Integrability preserving boundary conditions for this extension can be found as well.^[26]

Physical applications

The sine-Gordon model arises as the continuum limit of the Frenkel–Kontorova model which models crystal dislocations.

Dynamics in long Josephson junctions are well-described by the sine-Gordon equations, and conversely provide a useful experimental system for studying the sine-Gordon model.^[27]

The sine-Gordon model is in the same universality class as the effective action for a Coulomb gas of vortices and anti-vortices in the continuous classical XY model, which is a model of magnetism.^[28][29] The Kosterlitz–Thouless transition for vortices can therefore be derived from a renormalization group analysis of the sine-Gordon field theory.^[30][31]

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.^[32]