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

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

where $v_{tt}$ and $v_{xx}$ are Partial Derivatives. The equation can be transformed by defining

$$\xi\equiv {\textstyle{1\over 2}}(x-t)$$ (2)

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

giving

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

Traveling wave analysis gives

$$z-z_0=\sqrt{c^2-1} \int{df \over \sqrt{2[d-2\sin^2({\textstyle{1\over 2}}f)]}}.$$ (5)

For $d=0$,

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

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

Letting $z\equiv \xi\eta$ then gives

$$zf''+f'=\sin f.$$ (8)

Letting $g\equiv e^{if}$ gives

$$g''-{g'^2\over f}+{2g'-g^2+1\over 2z} = 0,$$ (9)

which is the third Painlevé Transcendent. Look for a solution of the form

$$v(x,t)=4\tan^{-1}\left[{\phi(x)\over \psi(t)}\right].$$ (10)

Taking the partial derivatives gives

$$\phi_{xx} = -k^2\phi^4+m^2\phi^2+n^2$$ (11)

$$\psi_{tt} = k^2\psi^4+(m^2-1)\psi^2-n^2,$$ (12)

which can be solved in terms of Elliptic Functions. A single Soliton solution exists with $k=n=0$, $m>1$:

$$v=4\tan^{-1}\left[{\mathop{\rm exp}\nolimits \left({\pm x-\beta t\over \sqrt{1-\beta^2}}\right)}\right],$$ (13)

where

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

A two-Soliton solution exists with $k=0$, $m>1$:

$$v=4\tan^{-1}\left[{\sinh(\beta mx)\over \beta\cosh(\beta mt)}\right].$$ (15)

A Soliton-antisoliton solution exists with $k\not=0$, $n=0$, $m^2>1$:

$$v=-4\tan^{-1}\left[{\sinh(\beta mx)\over\beta\cosh(mt)}\right].$$ (16)

A ``breather'' solution is

$$v=-4\tan^{-1}\left[{{m\over\sqrt{1-m^2}} {\sin(\sqrt{1-m^2t}\,)\over\cosh(mx)}}\right].$$ (17)

References

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