I have the following PDE in two dimensions
$$ 2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0, $$
with $u=u(x,y)$ on some domain of the plane. Now, numerically I can obtain the solutions very quickly specifying some domain and an initial Cauchy line (as the equation hyperbolic), but I wish to have a deeper understanding of the solutions, so I'd like to see if there's a way to obtain analytic solutions. For instance, I know that $u=\cos(2\arctan(y/x))$ and $u=\cos(2(\arctan(y/x))+\arccos(1+1/(x^2+y^2)))$ are analytic, particular solutions, which strongly suggests that some general solution with arbitrary constants is plausible.
Any ideas ? Have you seen this equation or someone similar before ?
Thank you some much.