“Sine-Gordon Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Sine-Gordon Equation

释义

Sine-Gordon Equation

A 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

			(11)
			(12)

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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