sine-Gordon equation

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The sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. 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 curvature –1 in 3-space,[1] and rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations.[2] This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions.

Origin of the equation and its name[edit]

There are two equivalent forms of the sine–Gordon equation. In the (real) space-time coordinates, denoted (xt), the equation reads:[3]

\, \varphi_{tt}- \varphi_{xx} + \sin\varphi = 0.

Passing to the light cone coordinates (uv), akin to asymptotic coordinates where

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

the equation takes the form:[4]

\varphi_{uv} = \sin\varphi.\,

This is the original form of the sine–Gordon equation, as it was considered in the nineteenth century in the course of investigation of surfaces of constant Gaussian curvature K = −1, also called pseudospherical surfaces. Choose a coordinate system for such a surface in which the coordinate mesh u = constant, v = constant is given by the asymptotic lines parameterized with respect to the arc length. The first fundamental form of the surface in these coordinates has a special form

 ds^2 = du^2 + 2\cos\varphi \,du\, dv + dv^2,\,

where  \varphi expresses the angle between the asymptotic lines, and for the second fundamental form, L = N = 0. Then the Codazzi-Mainardi equation expressing a compatibility condition between the first and second fundamental forms results in the sine–Gordon equation. 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.

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

 \varphi_{tt}- \varphi_{xx} + \varphi\ = 0.\,

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

\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,

\cos(\varphi) = \sum_{n=0}^\infty \frac{(-\varphi ^2)^n}{(2n)!},

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


\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}

Soliton solutions[edit]

An interesting feature of the sine–Gordon equation is the existence of soliton and multisoliton solutions.

1-soliton solutions[edit]

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

\varphi_\text{soliton}(x, t) := 4 \arctan e^{m \gamma (x - v t) + \delta}\,

where

\gamma^2 = \frac{1}{1 - v^2}.

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

\, \varphi_{tt}- \varphi_{xx} + m^2 \sin\varphi = 0.

The 1-soliton solution for which we have chosen the positive root for \gamma is called a kink, and represents a twist in the variable \varphi which takes the system from one solution \varphi=0 to an adjacent with \varphi=2\pi. The states \varphi=0(\textrm{mod}2\pi) 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 \gamma is called an antikink. The form of the 1-soliton solutions can be obtained through application of a Bäcklund transform to the trivial (constant vacuum) solution and the integration of the resulting first-order differentials:

{\varphi^\prime}_u = \varphi_u + 2\beta\sin\left(\frac{\varphi^\prime + \varphi}{2}\right),
{\varphi^\prime}_v = -\varphi_v + \frac{2}{\beta} \sin\left(\frac{\varphi^\prime - \varphi}{2}\right)\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 as discussed by Dodd and co-workers.[5] Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge \vartheta_{\textrm{K}}=-1. The alternative counterclockwise (right-handed) twist with topological charge \vartheta_{\textrm{AK}}=+1 will be an antikink.

Traveling kink soliton represents propagating clockwise twist.[6][7]
Traveling antikink soliton represents propagating counterclockwise twist.[6][7]

2-soliton solutions[edit]

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 kind of interaction is called an elastic collision.

Antikink-kink collision.[6][7]
Kink-kink collision.[6][7]

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]

Standing breather is a swinging in time coupled kink-antikink soliton.[6][7]
Large amplitude moving breather.[6][7]
Small-amplitude moving breather — looks exotic but essentially has a breather envelope.[6][7]

3-soliton solutions[edit]

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 \Delta_{\textrm{B}} is given by:

\Delta_B =\frac{2\textrm{arctanh}\sqrt{(1-\omega^{2})(1-v_\text{K}^2)}}{\sqrt{1-\omega^{2}}}

where v_\text{K} is the velocity of the kink, and \omega is the breather's frequency.[9] If the old position of the standing breather is x_{0}, after the collision the new position will be x_0 + \Delta_\text{B}.

Moving kink-standing breather collision.[6][7]
Moving antikink-standing breather collision.[6][7]

Related equations[edit]

The sinh–Gordon equation is given by

\varphi_{xx}- \varphi_{tt} = \sinh\varphi.\,

This is the Euler–Lagrange equation of the Lagrangian

\mathcal{L}={1\over 2}(\varphi_t^2 - \varphi_x^2) - \cosh\varphi.\,

Another closely related equation is the elliptic sine–Gordon equation, given by

\varphi_{xx} + \varphi_{yy} = \sin\varphi,\,

where \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.

A generalization is given by Toda field theory.

Quantum version[edit]

In quantum field theory the sine–Gordon model contains a parameter, it can be identified with Planck constant. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of breathers. The number of the breathers depends on the value of the parameter. Multi particle productions cancels on mass shell. Vanishing of two into four amplitude was explicitly checked in one loop approximation.

Semi-classical quantization of the sine–Gordon model was done by Ludwig Faddeev and Vladimir Korepin.[10] The exact quantum scattering matrix was discovered by Alexander Zamolodchikov. This model is S-dual to the Thirring model.

In finite volume and on a half line[edit]

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. On a half line the spectrum contains boundary bound states in addition to the solitons and breathers.

Supersymmetric sine–Gordon model[edit]

A supersymmetric extension of the sine–Gordon model also exists. Integrability preserving boundary conditions for this extension can be found as well.

See also[edit]

References[edit]

  1. ^ Bour E (1862). "Théorie de la déformation des surfaces". J. Ecole Imperiale Polytechnique 19: 1–48. 
  2. ^ Frenkel J, Kontorova T (1939). "On the theory of plastic deformation and twinning". Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya 1: 137–149. 
  3. ^ Rajaraman, R. (1989). Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. North-Holland Personal Library 15. North-Holland. pp. 34–45. ISBN 978-0-444-87047-6. 
  4. ^ Polyanin, Andrei D.; Valentin F. Zaitsev (2004). Handbook of Nonlinear Partial Differential Equations. Chapman & Hall/CRC Press. pp. 470–492. ISBN 978-1-58488-355-5. 
  5. ^ Dodd, Roger K.; J. C. Eilbeck; J. D. Gibbon; H. C. Morris (1982). Solitons and Nonlinear Wave Equations. London: Academic Press. ISBN 978-0-12-219122-0. 
  6. ^ a b c d e f g h i Georgiev DD, Papaioanou SN, Glazebrook JF (2004). "Neuronic system inside neurons: molecular biology and biophysics of neuronal microtubules". Biomedical Reviews 15: 67–75. doi:10.14748/bmr.v15.103. 
  7. ^ a b c d e f g h i Georgiev DD, Papaioanou SN, Glazebrook JF (2007). "Solitonic effects of the local electromagnetic field on neuronal microtubules". Neuroquantology 5 (3): 276–291. doi:10.14704/nq.2007.5.3.137. 
  8. ^ Rogers, C.; W. K. Schief (2002). Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. New York: Cambridge University Press. ISBN 978-0-521-01288-1. 
  9. ^ a b Miroshnichenko A, Vasiliev A, Dmitriev S. Solitons and Soliton Collisions.
  10. ^ Faddeev LD, Korepin VE (1978). "Quantum theory of solitons". Physics Reports 42 (1): 1–87. Bibcode:1978PhR....42....1F. doi:10.1016/0370-1573(78)90058-3. 

External links[edit]