Sources
Help build the largest human-edited directory on the web.
Submit a site - Become an editor
Wikipedia. Licensed under the GNU Free Documentation License.
Are you an expert in this subject? Join the discussion and share your knowledge at Wikipedia.org.
Sine-Gordon equation
From Wikipedia, the free encyclopedia
The sine-Gordon equation is a partial differential equation in two dimensions.[1][2] For a function 
of two real variables, x and t, it is
- φtt − φxx + sinφ = 0.
Contents |
The name is a pun on the Klein-Gordon equation, which is
The sine-Gordon equation is the Euler-Lagrange equation of the Lagrangian
If you Taylor-expand the cosine
and put this into the Lagrangian you get the Klein-Gordon Lagrangian plus some higher order terms
The sine-Gordon equation has the following 1-soliton solutions:
![\phi_{\mathrm{soliton}}(x, t) := 4 \arctan \exp[m \gamma (Xx - v t) + \delta]\,](http://upload.wikimedia.org/math/c/1/4/c144172f58f5884ae8d290a7e18112ce.png)
where 
The 1-soliton solution for which we have chosen the positive root for γ is called a kink, and represents a twist in the variable φ which takes the system from one solution φ = 0 to an adjacent with φ = 2π. The states φ = 0(mod2π) 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 γ is called an antikink.
The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model as discussed by Dodd and co-workers.[3] Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge Failed to parse (unknown function\textrmXtextrm): \vartheta_{\textrmXtextrm{AK}}=+1 will be an antikink.
Traveling kink soliton represents propagating clockwise twist.
|
Traveling antikink soliton represents propagating counterclockwise twist.
|
Multi-soliton solutions can be obtained with the Bäcklund transform. 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.
|
Kink-kink 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.[4]
Standing breather is a swinging in time coupled kink-antikink soliton.
|
Large amplitude moving breather.
|
Small amplitude moving breather - looks exotically but essentially has a breather envelope.
|
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 ΔB is given by:
where vK is the velocity of the kink, and ω is the breather's frequency.[4] If the old position of the standing breather is x0, after the collision the new position will be x0 + ΔB.
Moving kink-standing breather collision.
|
Moving antikink-standing breather collision.
|
In the above discussion, the sine-Gordon equation was expressed in space-time coordinates (x,t). If instead light cone coordinates are used, it takes the form:
where φ is a function of the two real variables u and v.
The formulation is better known in the differential geometry of pseudospherical surfaces, which are surfaces of negative constant Gaussian curvature K = −1. If such surfaces are described using an asymptotic line parameterization by arc length, then this form of the equation is the Codazzi-Mainardi equation, i.e., the integrability condition, for them.
Bäcklund transforms in soliton theory have their origins in the study of this equation, and the associated transformations of pseudospherical surfaces, by Bianchi and Bäcklund in the late 19th century.
The sinh-Gordon equation is given by
This is the Euler-Lagrange equation of the Lagrangian
Another closely related equation is the elliptic sine-Gordon equation, given by
where φ 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.
In quantum field theory the sine-Gordon model contains a parameter. 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.
On 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.
A supersymmetric extension of the sine-Gordon model also exists. Integrability preserving boundary conditions for this extension can be found as well.
- ^ Polyanin AD, Zaitsev VF. Handbook of Nonlinear Partial Differential Equations. Chapman & Hall/CRC Press, Boca Raton, 2004.
- ^ Rajaraman R. Solitons and instantons. North-Holland Personal Library, 1989.
- ^ Dodd RK, EilbecXk JC, Gibbon JD, Morris HC. Solitons and Nonlinear Wave Equations. Academic Press, London, 1982.
- ^ a b Miroshnichenko A, Vasiliev A, Dmitriev S. Solitons and Soliton Collisions.
- Sine-Gordon Equation at EqWorld: The World of Mathematical Equations.
- Sinh-Gordon Equation at EqWorld: The World of Mathematical Equations.