Here are non-linear partial differential equations, where $f$ and $g$ are functions of $x,t$ :
$g^2 (\partial_{x}f)(\partial_{t}f) - (\partial_{x}g) (\partial_{t}g) = 0, \quad g^{2}(\partial_{x}f)^{2} - (\partial_{x}g)^{2} = -1 , \quad g^{2}(\partial_{t}f)^{2} - (\partial_{t}g)^{2} = 1 .$
A "simple" solution is known , in fact the inverse expression of $x,t$ in function of $f,g$ is even "simpler".
Now, I am looking to the more logic, the more rigorous, and the most simple demonstration, which allows to find this "simple" solution.