# Sine-Gordon equation

[Contributions to this section are sorely needed!]

The sine-Gordon equation

${\displaystyle \Box u=\sin(u)}$

in ${\displaystyle R^{1+1}}$ arises in the study of optical pulses, or from the Scott model of a continuum of pendula hanging from a wire. It is a completely integrable equation, and has many interesting solutions, including "breather" solutions.

Because the non-linearity is bounded, GWP is easily obtained for ${\displaystyle L^{2}}$ or even ${\displaystyle L^{1}}$ data.

It is also closely related to the sinh-Gordon equation

${\displaystyle \Box u=\sinh(u)}$

and to Liouville's equation.