Thesine-Gordon equation is a second-ordernonlinear partial differential equation for a function${\displaystyle \phi }$ dependent on two variables typically denoted${\displaystyle x}$ and${\displaystyle t}$, involving thewave operator and thesine of${\displaystyle \phi }$.

It was originally introduced byEdmond Bour (1862) in the course of study ofsurfaces of constant negative curvature as theGauss–Codazzi equation for surfaces of constantGaussian curvature −1 in3-dimensional space.^[1] The equation was rediscovered byYakov Frenkel and Tatyana Kontorova (1939) in their study ofcrystal dislocations known as theFrenkel–Kontorova model.^[2]

This equation attracted a lot of attention in the 1970s due to the presence ofsoliton solutions,^[3] and is an example of anintegrable PDE. Among well-known integrable PDEs, the sine-Gordon equation is the onlyrelativistic system due to itsLorentz invariance.

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]

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

where partial derivatives are denoted by subscripts. Passing to thelight-cone coordinates (u, v), akin toasymptotic coordinates where

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

the equation takes the form^[5]

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

This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation ofsurfaces of constantGaussian curvatureK = −1, also calledpseudospherical surfaces.

Consider an arbitrary pseudospherical surface. Across every point on the surface there are twoasymptotic curves. This allows us to construct a distinguished coordinate system for such a surface, in whichu = constant,v = constant are the asymptotic lines, and the coordinates are incremented by thearc length on the surface. At every point on the surface, let${\displaystyle \phi }$ be the angle between the asymptotic lines.

Thefirst fundamental form of the surface is

$${\displaystyle d{s}^2=d{u}^2+2\cos \phi dudv+d{v}^2,}$$

and thesecond fundamental form is$${\displaystyle L=N=0,M=\sin \phi }$$and theGauss–Codazzi equation is$${\displaystyle {\phi }_{uv}=\sin \phi .}$$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 necessarilysingular due to theHilbert embedding theorem. In the simplest case,thepseudosphere, 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 torigid transformations. There is a theorem, sometimes called thefundamental 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.

The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century byBianchi andBäcklund led to the discovery ofBäcklund transformations. Another transformation of pseudospherical surfaces is theLie transform introduced bySophus Lie in 1879, which corresponds toLorentz 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${\displaystyle \phi }$ is a solution, then so is${\displaystyle \phi +2n\pi }$ for${\displaystyle n}$ an integer.

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${\displaystyle \phi }$, then schematically, the dynamics of the line of pendulum follows Newton's second law:$${\displaystyle \underset{\text{mass times acceleration}}{\underbrace{m{\phi }_{tt}}}=\underset{\text{tension}}{\underbrace{T{\phi }_{xx}}}-\underset{\text{gravity}}{\underbrace{mg\sin \phi }}}$$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${\displaystyle T{\phi }_{xx}}$, but more accurately${\displaystyle T{\phi }_{xx}(1+{\phi }_x^2{)}^{-3/2}}$. 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]

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

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

The sine-Gordon equation is theEuler–Lagrange equation of the field whoseLagrangian density is given by

$${\displaystyle {\mathcal{L}}_{\text{SG}}(\phi )=\frac{1}{2}({\phi }_t^2-{\phi }_x^2)-1+\cos \phi .}$$

Using theTaylor series expansion of thecosine in the Lagrangian,

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

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

An interesting feature of the sine-Gordon equation is the existence ofsoliton and multisoliton solutions.

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

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

where

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

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

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

The 1-soliton solution for which we have chosen the positive root for${\displaystyle \gamma }$ is called akink and represents a twist in the variable${\displaystyle \phi }$ which takes the system from one constant solution${\displaystyle \phi =0}$ to an adjacent constant solution${\displaystyle \phi =2\pi }$. The states${\displaystyle \phi \cong 2\pi n}$ 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 anantikink. The form of the 1-soliton solutions can be obtained through application of aBäcklund transform to the trivial (vacuum) solution and the integration of the resulting first-order differentials:

$${\displaystyle {\phi }_u^{\prime }={\phi }_u+2\beta \sin \frac{{\phi }^{\prime }+\phi }{2},}$$

$${\displaystyle {\phi }_v^{\prime }=-{\phi }_v+\frac{2}{\beta }\sin \frac{{\phi }^{\prime }-\phi }{2}\text{ with }\phi ={\phi }_0=0}$$

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.

Multi-soliton solutions can be obtained through continued application of theBäcklund transform to the 1-soliton solution, as prescribed by aBianchi 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 aphase shift. Since the colliding solitons recover theirvelocity andshape, such an interaction is called anelastic collision.

The kink-kink solution is given by$${\displaystyle {\phi }_{K/K}(x,t)=4\arctan \left(\frac{v\sinh \frac{x}{\sqrt{1-v^2}}}{\cosh \frac{vt}{\sqrt{1-v^2}}}\right)}$$

while the kink-antikink solution is given by$${\displaystyle {\phi }_{K/AK}(x,t)=4\arctan \left(\frac{v\cosh \frac{x}{\sqrt{1-v^2}}}{\sinh \frac{vt}{\sqrt{1-v^2}}}\right)}$$

Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as abreather. There are known three types of breathers:standing breather,traveling large-amplitude breather, andtraveling small-amplitude breather.^[11]

The standing breather solution is given by$${\displaystyle \phi (x,t)=4\arctan \left(\frac{\sqrt{1-{\omega }^2}\cos (\omega t)}{\omega \cosh (\sqrt{1-{\omega }^2}x)}\right).}$$

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

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

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${\displaystyle x_0+{\mathrm{\Delta }}_{\text{B}}}$.

Suppose that${\displaystyle \phi }$ is a solution of the sine-Gordon equation$${\displaystyle {\phi }_{uv}=\sin \phi .}$$

Then the systemwherea 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${\displaystyle \phi }$ 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${\displaystyle \phi }$ is the trivial solution${\displaystyle \phi \equiv 0}$, then${\displaystyle \psi }$ is the one-soliton solution with${\displaystyle a}$ related to the boost applied to the soliton.

Thetopological charge orwinding number of a solution${\displaystyle \phi }$ is$${\displaystyle N=\frac{1}{2\pi }{\int }_{\mathbb{R}}d\phi =\frac{1}{2\pi }\left[\phi (x=\mathrm{\infty },t)-\phi (x=-\mathrm{\infty },t)\right].}$$Theenergy of a solution${\displaystyle \phi }$ is$${\displaystyle E={\int }_{\mathbb{R}}dx\left(\frac{1}{2}({\phi }_t^2+{\phi }_x^2)+m^2(1-\cos \phi )\right)}$$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${\displaystyle N=0}$.

The sine-Gordon equation is equivalent to thecurvature of a particular${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$-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 bywhere the${\displaystyle {\sigma }_i}$ are thePauli matrices.Then the zero-curvature equation$${\displaystyle {\mathrm{\partial }}_v{A}_u-{\mathrm{\partial }}_u{A}_v+[A_u,A_v]=0}$$

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

The pair of matrices${\displaystyle A_u}$ and${\displaystyle A_v}$ are also known as aLax pair for the sine-Gordon equation, in the sense that the zero-curvature equation recovers the PDE rather than them satisfying Lax's equation.

Thesinh-Gordon equation is given by^[13]

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

This is theEuler–Lagrange equation of theLagrangian

$${\displaystyle \mathcal{L}=\frac{1}{2}({\phi }_t^2-{\phi }_x^2)-\cosh \phi .}$$

Another closely related equation is theelliptic sine-Gordon equation orEuclidean sine-Gordon equation, given by

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

where${\displaystyle \phi }$ is now a function of the variablesx andy. This is no longer a soliton equation, but it has many similar properties, as it is related to the sine-Gordon equation by theanalytic continuation (orWick rotation)y = it.

Theelliptic sinh-Gordon equation may be defined in a similar way.

Another similar equation comes from the Euler–Lagrange equation forLiouville field theory

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

A generalization is given byToda field theory.^[14] More precisely, Liouville field theory is the Toda field theory for the finiteKac–Moody algebra${\displaystyle {\mathfrak{s}\mathfrak{l}}_2}$, while sin(h)-Gordon is the Toda field theory for theaffine Kac–Moody algebra${\displaystyle {\widehat{\mathfrak{s}\mathfrak{l}}}_2}$.

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 containsboundary bound states in addition to the solitons and breathers.^[16]

Inquantum field theory the sine-Gordon model contains a parameter that can be identified with thePlanck constant. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number ofbreathers.^[17][18][19] 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 byLudwig Faddeev andVladimir Korepin.^[20] The exact quantumscattering matrix was discovered byAlexander Zamolodchikov.^[21]This model isS-dual to theThirring model, as discovered byColeman.^[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 underrenormalization: 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 afree massiveDirac field theory.

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

$${\displaystyle {\mathcal{L}}_{QsG}=\frac{1}{2}{\mathrm{\partial }}_{\mu }\phi {\mathrm{\partial }}^{\mu }\phi +\frac{1}{2}{m}_0^2{\phi }^2-\alpha (V_{\beta }+V_{-\beta })}$$

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

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

Thefinite regime is, where nocounterterms are needed to render the theory well-posed. Thesuper-renormalizable regime is, 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 anaffine sl₂ subalgebra, and the theory becomes strictly renormalizable (renormalizable, but not super-renormalizable).

Thestochastic ordynamical sine-Gordon model has been studied byMartin Hairer and Hao Shen^[25]allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.

The equation is$${\displaystyle {\mathrm{\partial }}_{t}u=\frac{1}{2}\mathrm{\Delta }u+c\sin (\beta u+\theta )+\xi ,}$$where${\displaystyle c,\beta ,\theta }$ are real-valued constants, and${\displaystyle \xi }$ is space-timewhite noise. The space dimension is fixed to 2. In the proof of existence of solutions, the thresholds${\displaystyle {\beta }^2=\frac{n}{n+1}8\pi }$ again play a role in determining convergence of certain terms.

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

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

Dynamics inlong 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 sameuniversality class as theeffective action for aCoulomb gas ofvortices and anti-vortices in the continuousclassical XY model, which is a model of magnetism.^[28][29] TheKosterlitz–Thouless transition for vortices can therefore be derived from arenormalization 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, thequantum Heisenberg model, in particular the XXZ model.^[32]

• Josephson effect
• Fluxon
• Shape waves

• sine-Gordon equation at EqWorld: The World of Mathematical Equations.
• Sinh-Gordon Equation at EqWorld: The World of Mathematical Equations.
• sine-Gordon equationArchived 2012-03-16 at theWayback Machine at NEQwiki, the nonlinear equations encyclopedia.

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