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The 'sine-Gordon equation' is a partial differential equation in two dimensions.^[1][2] For a function $phi$ of two real variables, ''x'' and ''t'', it is
:$(Box\; +\; sin)phi\; =\; phi\_\{tt\}-\; phi\_\{xx\}\; +\; sinphi\; =\; 0.$

Contents

Origin of the equation and name

1-soliton solutions

2-soliton solutions

3-soliton solutions

Relation with pseudospherical surfaces

Related equations

References and notes

External links

Origin of the equation and name

The name is a pun on the Klein-Gordon equation, which is
:$(Box\; +\; 1)phi\; =\; phi\_\{tt\}-\; phi\_\{xx\}\; +\; phi\; =\; 0.$
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
:
:::::::

1-soliton solutions

The sine-Gordon equation has the following 1-soliton solutions:
:
where
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 $phi$ which takes the system from one solution $phi=0$ to an adjacent with $phi=2pi$. The states $phi=0(\; extrm\{mod\}2pi)$ 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 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 . The alternative counterclockwise (right-handed) twist with topological charge will be an antikink.

Traveling ''kink'' soliton represents propagating clockwise twist.

Traveling ''antikink'' soliton represents propagating counterclockwise twist.

2-soliton solutions

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''.Miroshnichenko A, Vasiliev A, Dmitriev S. ''Solitons and Soliton Collisions.''

''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 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 $Delta\_\{\; extrm\{B\}\}$ is given by:

where $v\_\{\; extrm\{K\}\}$ 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 $x\_\{0\}+Delta\_\{\; extrm\{B\}\}$.

''Moving kink-standing breather'' collision.

''Moving antikink-standing breather'' collision.

Relation with pseudospherical surfaces

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.

Related equations

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'' = i''t''.
The 'elliptic sinh-Gordon equation' may be defined in a similar way.

References and notes

1. Polyanin AD, Zaitsev VF. ''Handbook of Nonlinear Partial Differential Equations''. Chapman & Hall/CRC Press, Boca Raton, 2004.
2. Rajaraman R. ''Solitons and instantons''. North-Holland Personal Library, 1989.
3. Dodd RK, Eilbeck JC, Gibbon JD, Morris HC. ''Solitons and Nonlinear Wave Equations''. Academic Press, London, 1982.

External links

★ Sine-Gordon Equation at EqWorld: The World of Mathematical Equations.

★ Sinh-Gordon Equation at EqWorld: The World of Mathematical Equations.