Sine-Gordon 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 of <math>\varphi</math>.

It was originally introduced by 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. The equation was rediscovered by in their study of crystal dislocations known as the Frenkel–Kontorova model.

This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions, 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 <math>(x,t)</math>, the equation reads:

: <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

the equation takes the form

<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 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 <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 due to the 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 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.

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

A mechanical model

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.

Naming

The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics: Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge <math>\theta_\text{K} = -1</math>. The alternative counterclockwise (right-handed) twist with topological charge <math>\theta_\text{AK} = +1</math> will be an antikink.

| frame|Traveling kink soliton represents a propagating clockwise twist.

| frame|Traveling antikink soliton represents a propagating counterclockwise twist. 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.

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.

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.

The is given by

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

This is the Euler–Lagrange equation of the 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 = 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

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

A generalization is given by Toda field theory. 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, on a line segment, or on a half line. It is possible to find boundary conditions which preserve the integrability of the model. 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. The exact quantum scattering matrix was discovered by Alexander Zamolodchikov.

This model is S-dual to the Thirring model, as discovered by Coleman. 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 field theory.

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

<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 renormalizability properties of the sine-Gordon theory change. The identification of these regimes is attributed to Jürg Fröhlich.

The finite regime is <math>\beta^2 < 4\pi</math>, where no counterterms 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. For <math>\beta^2 > 8\pi</math>, the theory becomes ill-defined . 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 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

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. Integrability preserving boundary conditions for this extension can be found as well.

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. The Kosterlitz–Thouless transition for vortices can therefore be derived from a renormalization group analysis of the sine-Gordon field theory.

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.

References

External links

sine-Gordon equation at EqWorld: The World of Mathematical Equations.
Sinh-Gordon Equation at EqWorld: The World of Mathematical Equations.
sine-Gordon equation at NEQwiki, the nonlinear equations encyclopedia.