TITLE: Sign-Gordon Equation
QUESTION [5 upvotes]: What can be said and done about the "SIGN-Gordon equation"?
$$\varphi_{tt}- \varphi_{xx} + \text{sgn}(\varphi) = 0.$$ 
It came up here.

REPLY [10 votes]: Turns out it appears in literature as the "signum-Gordon equation". For example the paper 
Signum-Gordon wave equation and its self-similar solutions