The sine-Gordon equation is a partial differential equation which appears in differential geometry and relativistic field theory. Its name is a wordplay on its similar form to the Klein-Gordon equation. The equation, as well as several solution techniques, were known in the 19th century, but the equation grew greatly in importance when it was realized that it led to solutions ("kink" and "antikink") with the collisional properties of solitons (Perring and Skyrme 1962; Tabor 1989, p. 307). The sine-Gordon equation also appears in a number of other physical applications (Barone 1971; Gibbon et al. 1979; Bishop and Schneider 1981; Davydov 1985; Infeld and Rowlands 2000, pp. 202 and 240), including the propagation of fluxons in Josephson junctions (a junction between two superconductors), the motion of rigid pendula attached to a stretched wire, and dislocations in crystals.
The sine-Gordon equation is
 |
(1)
|
where 
and 
are partial derivatives (Infeld and Rowlands
2000, p. 199).
The so-called double sine-Gordon equation is given by
![u_(xt)+/-[sinu+etasin(1/2u)]=0](/images/equations/Sine-GordonEquation/NumberedEquation2.svg) |
(2)
|
(Calogero and Degasperis 1982, p. 135; Zwillinger 1995, p. 135).
The equation can be transformed by defining
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(3)
|
 |  |  |
(4)
|
Then, by the chain rule,
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
This gives
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
Plugging in gives
 |
(13)
|
Another solution to the sine-Gordon equation is given by making the substitution 
, where 
, giving the ordinary
differential equation
 |
(14)
|
However, this cannot be solved analytically, since letting 
gives
 |
(15)
|
which is the third Painlevé transcendent (Tabor 1989, p. 309).
While the equation cannot be solved in all generality, several classes of solutions can be found by making the ansatz
![v(x,t)=4tan^(-1)[(phi(x))/(psi(t))].](/images/equations/Sine-GordonEquation/NumberedEquation6.svg) |
(16)
|
The reciprocal forms of the ansatz are related by the identity
 |
(17)
|
where 
. Replacing the quotient by its reciprocal therefore
corresponds to 
modulo 
, which preserves the sine-Gordon equation. The factor
of 4 is useful because 
is a rational
function of 
.
Plugging the ansatz (16) into the Sine-Gordon equation (1) then gives
 |
(18)
|
(Lamb 1980; Infeld and Rowlands 2000, pp. 199-200, typos corrected). Since the right-hand side contains two terms, one dependent only on 
and one only on 
, it can be eliminated by differentiating
both sides with respect to both 
and 
. Doing this and dividing the result
by 
, where these factors are nonzero, gives
 |
(19)
|
which can be written in the slightly simpler form
 |
(20)
|
Since the left term depends on 
only and the right term depends on 
only, separation of
variables can be used. For the solution family considered below, choose the separation
constant to be 
with 
. Then
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(21)
|
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(22)
|
Integrating gives
 |  |  |
(23)
|
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(24)
|
where 
and 
are constants of integration. Multiplying
through and integrating once more gives
 |  |  |
(25)
|
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(26)
|
where 
and 
are additional constants of integration.
The choices 
and 
satisfy equation (18). Taking 
and 
with 
and 
gives the compatible first
integrals
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(27)
|
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(28)
|
These equations can be integrated in terms of incomplete elliptic integrals of the first kind 
, but interesting classes
of solutions can be obtained by choosing simple values of the integration constants.
A single-soliton solution is obtained by taking 
and 
, in which case the equations have the solutions
 |  |  |
(29)
|
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(30)
|
Plugging into equation (16) then gives
 |  |  |
(31)
|
 |  | ![4tan^(-1){exp[m(x-(sqrt(m^2-1))/mt)]}](/images/equations/Sine-GordonEquation/Inline98.svg) |
(32)
|
 |  | ![4tan^(-1)[exp((x-betat)/(sqrt(1-beta^2)))],](/images/equations/Sine-GordonEquation/Inline101.svg) |
(33)
|
where 
has been defined as
 |
(34)
|
If 
had instead been defined with a minus sign, the same solution but with 
instead of 
would have been obtained. The positive solution is a soliton
also known as the "kink solution," while the negative solution is an antisoliton also known as the "antikink solution"
(Tabor 1989, pp. 306-307; Infeld and Rowlands 2000, p. 200).
A two-soliton solution exists with 
, 
, and 
:
![v=4tan^(-1)[(betasinh(mx))/(cosh(betamt))]](/images/equations/Sine-GordonEquation/NumberedEquation12.svg) |
(35)
|
(Infeld and Rowlands 2000, p. 200).
For 
, the same two-kink solution can be written
as
![v=4tan^(-1)[(betasinh(x/(sqrt(1-beta^2))))/(cosh((betat)/(sqrt(1-beta^2))))]](/images/equations/Sine-GordonEquation/NumberedEquation13.svg) |
(36)
|
(Perring and Skyrme 1962; Drazin and Johnson 1988; Tabor 1989, pp. 307-308).
A breather solution occurs for 
, 
, 
:
![v=-4tan^(-1)[m/(sqrt(1-m^2))(sin(sqrt(1-m^2)t))/(cosh(mx))].](/images/equations/Sine-GordonEquation/NumberedEquation14.svg) |
(37)
|
For fixed 
, this is a periodic function of 
with period 
(Infeld and Rowlands 2000, pp. 200-201).
