The sine-Gordon equation is a pun on the famous Klein-Gordon equation for relativistic quantum mechanics. The sine-Gordon equation in 1+1 dimensions, that is one spatial and one temporal, is:
$\frac{{\partial}^{2}\mathrm{Phi;}}{\partial {t}^{2}}-\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {x}^{2}}plus;\mathrm{sin}\left(\mathrm{Phi;}\right)equals;0$,
which is very similar to the 1+1 dimensional Klein-Gordon equation:
$\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {t}^{2}}-\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {x}^{2}}plus;\mathrm{Phi;}equals;0$.
The solutions to the sine-Gordon equation correspond to pseudospherical surfaces, where the function ${\mathrm{\Φ}}_{}\left(x\,t\right)$ is re-expressed in asymptote coordinates to describe a surface. Physicists instead interpret $\mathrm{\Φ}\left(x\,t\right)$ in real space-time as a wave propagating in the$\pm x-$direction through time $t$. This wave is nonlinear; consequently it does not dissipate when traveling at a constant velocity. One solution to the sine-Gordon equation is:
$\mathrm{\Φ}\left(x\,t\right)equals;4\mathrm{arctan}\left(\frac{\mathrm{cos}\mathrm{omega;}\sqrt{1-{\mathrm{omega;}}^{2}}}{\mathrm{omega;}\mathrm{cosh}\left(x\sqrt{1-{\mathrm{omega;}}^{2}}\right)}\right)$,
where $\mathrm{\ω}<1$is a periodic function (Nonlinear Waves, Solitons, and Chaos, 2nd ed. by Infeld and Rowlands). Two properties of this solution are
that it maintains its shape as it travels at constant velocity, and that its energy remains localized.
Turning this into asymptotic coordinates $u\,v$, gives the function:
${}$$\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$$f\left(u\,v\right)equals;4\mathrm{arctan}\left(\frac{b}{\sqrt{1-{b}^{2}}}\frac{\mathrm{sin}\left(\sqrt{1-{b}^{2}}v\right)}{\mathrm{cosh}\left(bu\right)}\right)$,
for the surface corresponding to a breather function. The parameter $b$ changes the number of 'ribs' in the surface, and as $b\to 1$, the surface transforms into the Kuen surface.