A Partial Differential Equation which appears in differential geometry and relativistic field theory. Its name is
a pun on its similar form to the KleinGordon Equation. The sineGordon 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 twoSoliton solution exists with , :

(15) 
A Solitonantisoliton 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. 199200, 1990.
© 19969 Eric W. Weisstein
19990526