Sine-Gordon Equation

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

where and are partial derivatives (Infeld and Rowlands 2000, p. 199).

The so-called double sine-Gordon equation is given by

(Calogero and Degasperis 1982, p. 135; Zwillinger 1995, p. 135).

The equation can be transformed by defining

Then, by the chain rule,

This gives

Plugging in gives

Another solution to the sine-Gordon equation is given by making the substitution , where , giving the ordinary differential equation

However, this cannot be solved analytically, since letting gives

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 that the solution is of of the form

This can be physically motivated on the grounds that the identity

means that interchanging space and time variables preserves the solution, as required by the symmetry of the sine-Gordon equation (1). (Although the reason for the factor of 4 is not entirely clear.)

Plugging the ansatz (16) into the Sine-Gordon equation (1) then gives

(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 side with respect to both and . Doing this and dividing the result by gives

which can be written in the slightly simpler form

Since the left term depends on only and the right term depends on only, separation of variables can be used to write

where the separation constant is assumed to be positive. Rewriting these two equations then gives

These can be integrated directly to give

where and are constants of integration which are connected through equation (◇). Clearing denominators,

The final form of the Sine-Gordon equation with the above separation is then

(Infeld and Rowlands 2000, p. 200, typos corrected), where is another constant of integration. These equations can be solved in general in terms of incomplete elliptic integrals of the first kind , but interesting classes of solution can be investigated by picking particularly simple values of the integration constants.

A single-soliton solution is obtained by taking and , in which case the equations have the solutions

Plugging into equation (◇) then gives

where has been defined as

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 , :

(Infeld and Rowlands 2000, p. 200).

A two-kink solution is given by

(Perring and Skyrme 1962; Drazin 1988; Tabor 1989, pp. 307-308).

A "breather" solution occurs for , , :

For a fixed , this is a periodic function of with frequency (Infeld and Rowlands 2000, pp. 200-201).