[Contributions to this section are sorely needed!]
The sine-Gordon equation
in 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 or even data.
It is also closely related to the sinh-Gordon equation
and to Liouville's equation.