What can be said and done about the "SIGN-Gordon equation"? $$\varphi_{tt}- \varphi_{xx} + \text{sgn}(\varphi) = 0.$$ It came up here.
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3$\begingroup$ 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. $\endgroup$– Steve HuntsmanCommented Apr 25, 2010 at 16:20
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2$\begingroup$ 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. $\endgroup$– Kaveh KhodjastehCommented Apr 25, 2010 at 17:10
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1$\begingroup$ Jim, Kaveh is talking about the sgn function, not the sin function. (That is, it's a pun on the sine-Gordon equation.) $\endgroup$– Qiaochu YuanCommented Apr 25, 2010 at 18:28
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$\begingroup$ 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. $\endgroup$– Jim HumphreysCommented Apr 25, 2010 at 19:23
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Turns out it appears in literature as the "signum-Gordon equation". For example the paper Signum-Gordon wave equation and its self-similar solutions
answered Apr 26, 2010 at 11:56