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Sine-Gordon Equation

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

v_{tt}-v_{xx}+\sin v=0,
\end{displaymath} (1)

where $v_{tt}$ and $v_{xx}$ are Partial Derivatives. The equation can be transformed by defining
\xi\equiv {\textstyle{1\over 2}}(x-t)
\end{displaymath} (2)

\eta\equiv {\textstyle{1\over 2}}(x+t),
\end{displaymath} (3)

v_{\xi\eta}=\sin v.
\end{displaymath} (4)

Traveling wave analysis gives
z-z_0=\sqrt{c^2-1} \int{df \over \sqrt{2[d-2\sin^2({\textstyle{1\over 2}}f)]}}.
\end{displaymath} (5)

For $d=0$,
z-z_0=\pm\sqrt{1-c^2}\,\ln[\pm \tan({\textstyle{1\over 4}}f)]
\end{displaymath} (6)

f(z)= \pm 4\tan^{-1} [e^{\pm (z-z_0)/(1-c^2)^{1/2}}].
\end{displaymath} (7)

Letting $z\equiv \xi\eta$ then gives
zf''+f'=\sin f.
\end{displaymath} (8)

Letting $g\equiv e^{if}$ gives
g''-{g'^2\over f}+{2g'-g^2+1\over 2z} = 0,
\end{displaymath} (9)

which is the third Painlevé Transcendent. Look for a solution of the form
v(x,t)=4\tan^{-1}\left[{\phi(x)\over \psi(t)}\right].
\end{displaymath} (10)

Taking the partial derivatives gives
$\displaystyle \phi_{xx}$ $\textstyle =$ $\displaystyle -k^2\phi^4+m^2\phi^2+n^2$ (11)
$\displaystyle \psi_{tt}$ $\textstyle =$ $\displaystyle k^2\psi^4+(m^2-1)\psi^2-n^2,$ (12)

which can be solved in terms of Elliptic Functions. A single Soliton solution exists with $k=n=0$, $m>1$:
v=4\tan^{-1}\left[{\mathop{\rm exp}\nolimits \left({\pm x-\beta t\over \sqrt{1-\beta^2}}\right)}\right],
\end{displaymath} (13)

\beta\equiv {\sqrt{m^2-1}\over m}.
\end{displaymath} (14)

A two-Soliton solution exists with $k=0$, $m>1$:
v=4\tan^{-1}\left[{\sinh(\beta mx)\over \beta\cosh(\beta mt)}\right].
\end{displaymath} (15)

A Soliton-antisoliton solution exists with $k\not=0$, $n=0$, $m^2>1$:
v=-4\tan^{-1}\left[{\sinh(\beta mx)\over\beta\cosh(mt)}\right].
\end{displaymath} (16)

A ``breather'' solution is
v=-4\tan^{-1}\left[{{m\over\sqrt{1-m^2}} {\sin(\sqrt{1-m^2t}\,)\over\cosh(mx)}}\right].
\end{displaymath} (17)


Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos. Cambridge, England: Cambridge University Press, pp. 199-200, 1990.

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© 1996-9 Eric W. Weisstein