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The sine-Gordon equation is a nonlinear hyperbolic
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
in 1 + 1 dimensions involving the d'Alembert operator and the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
of the unknown function. It was originally introduced by in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of curvature −1 in 3-space, and 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 In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
solutions.


Origin of the equation and its name

There are two equivalent forms of the sine-Gordon equation. In the ( real) ''space-time coordinates'', denoted (''x'', ''t''), the equation reads: : \varphi_ - \varphi_ + \sin\varphi = 0, where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (''u'', ''v''), akin to ''asymptotic coordinates'' where : u = \frac, \quad v = \frac, the equation takes the form : \varphi_ = \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 In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...
''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 transformation Backlund is a Swedish surname. Notable people with the surname include: * Albert Victor Bäcklund (1845-1922), mathematician * Bengt Backlund (1926–2006), Swedish flatwater canoer * Bob Backlund (born 1949), American professional wrestler * Fil ...
s. 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. The name "sine-Gordon equation" is a pun on the well-known
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...
in physics: : \varphi_ - \varphi_ + \varphi = 0. The sine-Gordon equation is the Euler–Lagrange equation of the field whose
Lagrangian density Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
is given by : \mathcal_\text(\varphi) = \frac (\varphi_t^2 - \varphi_x^2) - 1 + \cos\varphi. Using the Taylor series expansion of the cosine in the Lagrangian, : \cos(\varphi) = \sum_^\infty \frac, it can be rewritten as the Klein–Gordon Lagrangian plus higher-order terms: : \begin \mathcal_\text(\varphi) &= \frac (\varphi_t^2 - \varphi_x^2) - \frac + \sum_^\infty \frac \\ &= \mathcal_\text(\varphi) + \sum_^\infty \frac. \end


Soliton solutions

An interesting feature of the sine-Gordon equation is the existence of
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
and multisoliton solutions.


1-soliton solutions

The sine-Gordon equation has the following 1-
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
solutions: : \varphi_\text(x, t) := 4 \arctan \left(e^\right), where : \gamma^2 = \frac, and the slightly more general form of the equation is assumed: : \varphi_ - \varphi_ + 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 \pmod 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'_u = \varphi_u + 2\beta \sin\frac, : \varphi'_v = -\varphi_v + \frac \sin\frac \text \varphi = \varphi_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. Here we take a clockwise (
left-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subject ...
) twist of the elastic ribbon to be a kink with topological charge \theta_\text = -1. The alternative counterclockwise (
right-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
) twist with topological charge \theta_\text = +1 will be an antikink.


2-soliton solutions

Multi-
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
solutions can be obtained through continued application of the Bäcklund transform to the 1-soliton solution, as prescribed by a
Bianchi lattice Bianchi may refer to: Places *Bianchi, Calabria, a ''comune'' in the Province of Cosenza, Italy Manufacturing * Bianchi Bicycles (F.I.V. Edoardo Bianchi S.p.A.), an Italian manufacturer of bicycles, and former manufacturer of motorcycles and ...
relating the transformed results. 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 Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
and
shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...
, such kind of
interaction Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to: Science * Interaction hypothesis, a theory of second language acquisition * Interaction (statistics) * Interaction ...
is called an elastic collision. Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a ''
breather 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 ...
''. There are known three types of breathers: ''standing breather'', ''traveling large-amplitude breather'', and ''traveling small-amplitude breather''.Miroshnichenko A. E., Vasiliev A. A., Dmitriev S. V.
Solitons and Soliton Collisions
'.


3-soliton solutions

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_\text is given by : \Delta_\text =\frac, where v_\text is the velocity of the kink, and \omega is the breather's frequency. If the old position of the standing breather is x_0, after the collision the new position will be x_0 + \Delta_\text.


FDTD (1D) video simulation of a soliton with forces

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 the ''p''–''v'' field, which reaches the peer. # Solitons begin to move. # They meet in the middle and annihilate. # Mass is spread as wave.


Related equations

The is given by : \varphi_ - \varphi_ = \sinh\varphi. This is the Euler–Lagrange equation of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
: \mathcal = \frac (\varphi_t^2 - \varphi_x^2) - \cosh\varphi. Another closely related equation is the elliptic sine-Gordon equation, given by : \varphi_ + \varphi_ = \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 In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
(or
Wick rotation In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that su ...
) ''y'' = i''t''. 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 \varphi_ - \varphi_ = 2e^. A generalization is given by Toda field theory.


Quantum version

In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
the sine-Gordon model contains a parameter that can be identified with the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of
breather 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 ...
s. The number of the breathers depends on the value of the parameter. Multiparticle productions 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 Alexander Borisovich Zamolodchikov (russian: Алекса́ндр Бори́сович Замоло́дчиков; born September 18, 1952) is a Russian physicist, known for his contributions to condensed matter physics, two-dimensional conforma ...
. This model is S-dual to the Thirring model.


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


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.


See also

*
Josephson effect In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum me ...
* Fluxon * Shape waves


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. {{DEFAULTSORT:Sine-Gordon Equation Solitons Differential geometry Surfaces Exactly solvable models Equations of physics Mathematical physics Articles containing video clips