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
![\begin{displaymath}
v_{tt}-v_{xx}+\sin v=0,
\end{displaymath}](s1_1468.gif) |
(1) |
where
and
are Partial Derivatives.
The equation can be transformed by defining
![\begin{displaymath}
\xi\equiv {\textstyle{1\over 2}}(x-t)
\end{displaymath}](s1_1471.gif) |
(2) |
![\begin{displaymath}
\eta\equiv {\textstyle{1\over 2}}(x+t),
\end{displaymath}](s1_1472.gif) |
(3) |
giving
![\begin{displaymath}
v_{\xi\eta}=\sin v.
\end{displaymath}](s1_1473.gif) |
(4) |
Traveling wave analysis gives
![\begin{displaymath}
z-z_0=\sqrt{c^2-1} \int{df \over \sqrt{2[d-2\sin^2({\textstyle{1\over 2}}f)]}}.
\end{displaymath}](s1_1474.gif) |
(5) |
For
,
![\begin{displaymath}
z-z_0=\pm\sqrt{1-c^2}\,\ln[\pm \tan({\textstyle{1\over 4}}f)]
\end{displaymath}](s1_1476.gif) |
(6) |
![\begin{displaymath}
f(z)= \pm 4\tan^{-1} [e^{\pm (z-z_0)/(1-c^2)^{1/2}}].
\end{displaymath}](s1_1477.gif) |
(7) |
Letting
then gives
![\begin{displaymath}
zf''+f'=\sin f.
\end{displaymath}](s1_1479.gif) |
(8) |
Letting
gives
![\begin{displaymath}
g''-{g'^2\over f}+{2g'-g^2+1\over 2z} = 0,
\end{displaymath}](s1_1481.gif) |
(9) |
which is the third Painlevé Transcendent. Look for a solution of the form
![\begin{displaymath}
v(x,t)=4\tan^{-1}\left[{\phi(x)\over \psi(t)}\right].
\end{displaymath}](s1_1482.gif) |
(10) |
Taking the partial derivatives gives
which can be solved in terms of Elliptic Functions. A single Soliton solution
exists with
,
:
![\begin{displaymath}
v=4\tan^{-1}\left[{\mathop{\rm exp}\nolimits \left({\pm x-\beta t\over \sqrt{1-\beta^2}}\right)}\right],
\end{displaymath}](s1_1489.gif) |
(13) |
where
![\begin{displaymath}
\beta\equiv {\sqrt{m^2-1}\over m}.
\end{displaymath}](s1_1490.gif) |
(14) |
A two-Soliton solution exists with
,
:
![\begin{displaymath}
v=4\tan^{-1}\left[{\sinh(\beta mx)\over \beta\cosh(\beta mt)}\right].
\end{displaymath}](s1_1491.gif) |
(15) |
A Soliton-antisoliton solution exists with
,
,
:
![\begin{displaymath}
v=-4\tan^{-1}\left[{\sinh(\beta mx)\over\beta\cosh(mt)}\right].
\end{displaymath}](s1_1495.gif) |
(16) |
A ``breather'' solution is
![\begin{displaymath}
v=-4\tan^{-1}\left[{{m\over\sqrt{1-m^2}} {\sin(\sqrt{1-m^2t}\,)\over\cosh(mx)}}\right].
\end{displaymath}](s1_1496.gif) |
(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