The sine-Gordon equation is a second-order nonlinear partial differential equation for a function
\varphi
x
t
\varphi
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.[1] The equation was rediscovered by 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.
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
(x,t)
\varphitt-\varphixx+\sin\varphi=0,
where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (u, v), akin to asymptotic coordinates where
u=
x+t | |
2 |
, v=
x-t | |
2 |
,
the equation takes the form[5]
\varphiuv=\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
\varphi
The first fundamental form of the surface is
ds2=du2+2\cos\varphidudv+dv2,
and the second fundamental form isand the Gauss–Codazzi equation isThus, 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.
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
\varphi
\varphi+2n\pi
n
See main article: Frenkel–Kontorova 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
x
\varphi
Note that this is not exactly correct, since the net force on a pendulum due to the tension is not precisely
T\varphixx
T\varphixx
2) | |
(1+\varphi | |
x |
-3/2
The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics:[4]
\varphitt-\varphixx+\varphi=0.
The sine-Gordon equation is the Euler–Lagrange equation of the field whose Lagrangian density is given by
l{L}SG(\varphi)=
1 | |
2 |
2 | |
(\varphi | |
t |
-
2) | |
\varphi | |
x |
-1+\cos\varphi.
Using the Taylor series expansion of the cosine in the Lagrangian,
\cos(\varphi)=
infty | |
\sum | |
n=0 |
(-\varphi2)n | |
(2n)! |
,
it can be rewritten as the Klein–Gordon Lagrangian plus higher-order terms:
\begin{align} l{L}SG(\varphi)&=
1 | |
2 |
2 | |
(\varphi | |
t |
-
2) | |
\varphi | |
x |
-
\varphi2 | |
2 |
+
infty | |
\sum | |
n=2 |
(-\varphi2)n | |
(2n)! |
\\ &=l{L}KG(\varphi)+
infty | |
\sum | |
n=2 |
(-\varphi2)n | |
(2n)! |
. \end{align}
An interesting feature of the sine-Gordon equation is the existence of soliton and multisoliton solutions.
The sine-Gordon equation has the following 1-soliton solutions:
\varphisoliton(x,t):=4\arctan\left(em\right),
where
\gamma2=
1 | |
1-v2 |
,
and the slightly more general form of the equation is assumed:
\varphitt-\varphixx+m2\sin\varphi=0.
The 1-soliton solution for which we have chosen the positive root for
\gamma
\varphi
\varphi=0
\varphi=2\pi
\varphi\cong2\pin
\gamma
\varphi'u=\varphiu+2\beta\sin
\varphi'+\varphi | |
2 |
,
\varphi'v=-\varphiv+
2 | \sin | |
\beta |
\varphi'-\varphi | |
2 |
with\varphi=\varphi0=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.[7] Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge
\thetaK=-1
\thetaAK=+1
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.[8] 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
while the kink-antikink solution is given by
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.[9]
The standing breather solution is given by
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
\DeltaB
\DeltaB=
2\operatorname{artanh | |
\sqrt{(1 |
-\omega2)(1-
2)}}{\sqrt{1 | |
v | |
K |
-\omega2}},
where
vK
\omega
x0
x0+\DeltaB
See also: Bäcklund transform. Suppose that
\varphi
\varphiuv=\sin\varphi.
Then the system
\begin{align} \psiu&=\varphiu+2a\sinl(
\psi+\varphi | |
2 |
r)\\ \psiv&=-\varphiv+
2 | |
a |
\sinl(
\psi-\varphi | |
2 |
r) \end{align}
\psi
\varphi
\psi
By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.
For example, if
\varphi
\varphi\equiv0
\psi
a
The topological charge or winding number of a solution
\varphi
\varphi
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
N=0
The sine-Gordon equation is equivalent to the curvature of a particular
ak{su}(2)
R2
Explicitly, with coordinates
(u,v)
R2
A\mu
\sigmai
is equivalent to the sine-Gordon equation
\varphiuv=\sin\varphi
F\mu\nu=[\partial\mu-A\mu,\partial\nu-A\nu]
The pair of matrices
Au
Av
The is given by[11]
\varphixx-\varphitt=\sinh\varphi.
This is the Euler–Lagrange equation of the Lagrangian
l{L}=
1 | |
2 |
2 | |
(\varphi | |
t |
-
2) | |
\varphi | |
x |
-\cosh\varphi.
Another closely related equation is the elliptic sine-Gordon equation or Euclidean sine-Gordon equation, given by
\varphixx+\varphiyy=\sin\varphi,
where
\varphi
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
ak{sl}2
\hatak{sl}2
One can also consider the sine-Gordon model on a circle,[13] on a line segment, or on a half line.[14] It is possible to find boundary conditions which preserve the integrability of the model.[14] On a half line the spectrum contains boundary bound states in addition to the solitons and breathers.[14]
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.[15] [16] [17] 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.[18] The exact quantum scattering matrix was discovered by Alexander Zamolodchikov.[19] This model is S-dual to the Thirring model, as discovered by Coleman.[20] 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
\alpha0,\beta
\gamma0
\alpha0
\gamma0
\beta
\beta=\sqrt{4\pi}
The quantum sine-Gordon equation should be modified so the exponentials become vertex operators
l{L}QsG=
1 | |
2 |
\partial\mu\varphi\partial\mu\varphi+
1 | |
2 |
2\varphi | |
m | |
0 |
2-\alpha(V\beta+V-\beta)
with
V\beta=:ei\beta\varphi:
For different values of the parameter
\beta2
The finite regime is
\beta2<4\pi
4\pi<\beta2<8\pi
n | |
n+1 |
8\pi
\beta2>8\pi
\beta2=4\pi
\beta2=8\pi
The stochastic or dynamical sine-Gordon model has been studied by Martin Hairer and Hao Shen[23] allowing heuristic results from the quantum sine-Gordon theory to be proven in a statistical setting.
The equation iswhere
c,\beta,\theta
\xi
\beta2=
n | |
n+1 |
8\pi
A supersymmetric extension of the sine-Gordon model also exists.[24] Integrability preserving boundary conditions for this extension can be found as well.[24]
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.[25]
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.[26] [27] The Kosterlitz–Thouless transition for vortices can therefore be derived from a renormalization group analysis of the sine-Gordon field theory.[28] [29]
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.[30]