Sign-Gordon Equation

What can be said and done about the "SIGN-Gordon equation"? $$\varphi_{tt}- \varphi_{xx} + \text{sgn}(\varphi) = 0.$$ It came up here.

-
Rescale $x$, then rescale $t$ identically so the derivatives both get the same constant multipliers. The constants that pop out don't affect the signum, so you can rescale $\varphi$. So this is a kind of scaling limit of sinh-Gordon. –  Steve Huntsman Apr 25 '10 at 16:20
I thought about that but sinh would scale like O(\phi) in the small limit while sgn(\phi) scales like O(1). Thanks for the idea of scaling though. –  Kaveh Khodjasteh Apr 25 '10 at 17:10
Jim, Kaveh is talking about the sgn function, not the sin function. (That is, it's a pun on the sine-Gordon equation.) –  Qiaochu Yuan Apr 25 '10 at 18:28
Sorry, I was distracted by the header and didn't look far enough into the question. Some of my colleagues have been fond of sine-Gordon, but I'm an outsider. –  Jim Humphreys Apr 25 '10 at 19:23