释义 |
Sine-Gordon EquationA Partial Differential Equation which appears in differential geometry and relativistic field theory. Its name isa 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.
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