Sine-Gordon equation

Remarks

Example: the static kink $u(x,t) = 4 \arctan(e^x)$ is a solution. Put $\theta = \arctan(e^x)$; then $\sin 2\theta = \mathrm{sech}\, x$ and $\cos 2\theta = -\tanh x$. So $\sin u = \sin 4\theta = -2\, \mathrm{sech}\, x \tanh x$. Also $u_x = 2\, \mathrm{sech}\, x$, so $u_{xx} = -2\, \mathrm{sech}\, x \tanh x = \sin u$, and $u_{tt} = 0$. For $\abs{v} < 1$, the moving kink $4 \arctan\!\big(\exp\big((x - vt)/\sqrt{1 - v^2}\big)\big)$ is also a solution, by Lorentz invariance.

Non-example: $u(x,t) = x$ is not a solution, because $u_{tt} - u_{xx} + \sin u = \sin x$ is not identically $0$.

In the coordinates $\xi = (x+t)/2$ and $\eta = (x-t)/2$ the equation becomes $u_{\xi\eta} = \sin u$. This is the form in which it arises for surfaces of Gaussian curvature $-1$.