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 EdmondBour ( 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 FrenkelandKontorova ( 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.

- Origin of the equation and its name
- Soliton solutions
- 1-soliton solutions
- 2-soliton solutions
- 3-soliton solutions
- FDTD (1D) video simulation of a soliton with forces
- Related equations
- Quantum version
- In finite volume and on a half line
- Supersymmetric sine-Gordon model
- See also
- References
- External links

There are two equivalent forms of the sine-Gordon equation. In the (real) *space-time coordinates*, denoted (*x*, *t*), the equation reads:^{ [3] }

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:^{ [4] }

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

where 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. Another transformation of pseudospherical surfaces is the Lie transform introduced by Sophus Lie in 1879, which corresponds to Lorentz boosts in terms of light-cone coordinates, thus the sine-Gordon equation is Lorentz invariant.^{ [5] }

The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics:^{ [3] }

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

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

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

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:

where

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

The 1-soliton solution for which we have chosen the positive root for is called a *kink*, and represents a twist in the variable which takes the system from one solution to an adjacent with . The states 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 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:

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*.^{ [6] } Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge . The alternative counterclockwise (right-handed) twist with topological charge will be an antikink.

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.

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] }

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 is given by:

where is the velocity of the kink, and is the breather's frequency.^{ [9] } If the old position of the standing breather is , after the collision the new position will be .

The following video shows a simulation of two parking solitons. Both send out a pressure-speed field with different polarity. Because the end of the 1D space is not terminated symmetrically - waves are reflected.

Lines in the video:

- Cos() part of the soliton.
- Sin() part of the soliton.
- Angle acceleration of the soliton.
- Pressure-Component of the field with different polarity.
- Speed-Component of the field - direction dependent.

Steps:

- Solitons send out not bound energy as waves.
- Solitons send the p-v-field which reaches the peer.
- Solitons begin to move.
- They meet in the middle and annihilate.
- Mass is spread as wave.

The **sinh-Gordon equation** is given by^{ [10] }

This is the Euler–Lagrange equation of the Lagrangian

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

where 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* = i*t*.

The **elliptic sinh-Gordon equation** may be defined in a similar way.

A generalization is given by Toda field theory.^{ [11] }

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. 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.^{ [12] } The exact quantum scattering matrix was discovered by Alexander Zamolodchikov. This model is S-dual to the Thirring model.

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.

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

In physics and geometry, a **catenary** is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.

**Euler's formula**, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

In mechanics and physics, **simple harmonic motion** is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, the **Jacobi elliptic functions** are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance. They are found in the description of the motion of a pendulum, as well as in the design of the electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation *sn* for *sin*. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi (1829).

In mathematics, **Bäcklund transforms** or **Bäcklund transformations** relate partial differential equations and their solutions. They are an important tool in soliton theory and integrable systems. A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter. It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other.

In trigonometry, **tangent half-angle formulas** relate the tangent of half of an angle to trigonometric functions of the entire angle. Among these are the following

In theoretical physics, the (one-dimensional) **nonlinear Schrödinger equation** (**NLSE**) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In physics, a **breather** is a nonlinear wave in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for infinitesimal amplitudes, which tends towards an even distribution of initially localized energy.

In mathematics, the **Weierstrass–Enneper parameterization** of minimal surfaces is a classical piece of differential geometry.

**Toroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.

**Prolate spheroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

**Oblate spheroidal coordinates** are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the *x*-*y* plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.

In mathematics, the **sine** is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle. For an angle , the sine function is denoted simply as .

In mathematics, the **cylindrical harmonics** are a set of linearly independent functions that are solutions to Laplace's differential equation, , expressed in cylindrical coordinates, *ρ*, *φ*, and *z* (height). Each function *V*_{n}(*k*) is the product of three terms, each depending on one coordinate alone. The *ρ*-dependent term is given by Bessel functions.

In mathematical physics, the **Hunter–Saxton equation**

The **Frenkel–Kontorova model**, also known as the **FK model**, is a fundamental model of low-dimensional nonlinear physics.

- ↑ Bour E (1862). "Théorie de la déformation des surfaces".
*Journal de l'École Impériale Polytechnique*.**19**: 1–48. - ↑ Frenkel J, Kontorova T (1939). "On the theory of plastic deformation and twinning".
*Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya*.**1**: 137–149. - 1 2 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. - ↑ 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. - ↑ Terng, C. L., & Uhlenbeck, K. (2000). "Geometry of solitons" (PDF).
*Notices of AMS*.**47**(1): 17–25.CS1 maint: multiple names: authors list (link) - ↑ 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. - 1 2 3 4 5 6 7 8 9 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 . - ↑ 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. - 1 2 Miroshnichenko A, Vasiliev A, Dmitriev S.
*Solitons and Soliton Collisions.* - ↑ Polyanin, Andrei D.; Zaitsev, Valentin F.
*Handbook of Nonlinear Partial Differential Equations*(Second ed.). Boca Raton: CRC Press. p. 485. ISBN 978-1-4200-8723-9. - ↑ Yuanxi, Xie; Tang, Jiashi (February 2006). "A unified method for solving sinh-Gordon--type equations".
*Il Nuovo Cimento B*.**121**(2): 115–121. Bibcode:2006NCimB.121..115X. doi:10.1393/ncb/i2005-10164-6. - ↑ 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.

- 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.

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