A Partial Differential Equation which appears in differential geometry and relativistic field theory. Its name is
a pun on its similar form to the Klein-Gordon Equation. The sine-Gordon equation is
|
(1) |
where and are Partial Derivatives.
The equation can be transformed by defining
|
(2) |
|
(3) |
giving
|
(4) |
Traveling wave analysis gives
|
(5) |
For ,
|
(6) |
|
(7) |
Letting
then gives
|
(8) |
Letting
gives
|
(9) |
which is the third Painlevé Transcendent. Look for a solution of the form
|
(10) |
Taking the partial derivatives gives
which can be solved in terms of Elliptic Functions. A single Soliton solution
exists with , :
|
(13) |
where
|
(14) |
A two-Soliton solution exists with , :
|
(15) |
A Soliton-antisoliton solution exists with ,
, :
|
(16) |
A ``breather'' solution is
|
(17) |
References
Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos. Cambridge, England:
Cambridge University Press, pp. 199-200, 1990.
© 1996-9 Eric W. Weisstein
1999-05-26