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 $u=u(x,y)$ with values between $-1$ and $1$, or alternatively

$$ 2\partial_x\partial_y\sin2\theta(x,y)+\left(\partial^2_x-\partial^2_y \right)\cos2\theta(x,y)=0, $$

with real $\theta(x,y)\sim\theta(x,y)+2\pi$, 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 $\theta(x,y)=\arctan(y/x)\pm1/2\arccos(c_1+c_2/(x^2+y^2))$, with $c_1, c_2$ some reals constants, are analytic, particular solutions, which strongly suggests that some general solution with arbitrary constants is plausible.

The problem is encountered in the context of elasticity of thin sheets. A so-called director field is imprinted on a thin elastic sheet, and it generates curvature upon a process called activation [1]. The director field $\theta(x,y)$ will induce a Riemannian metric on the new, curved sheet

$$ g(x,y)=R[\theta(x,y)]diag(\lambda_1,\lambda_2)R[\theta(x,y)]^T, $$ with $R[\theta(x,y)]$ a $2\times2$ rotation matrix and $\lambda_1,\lambda_2$ some positive, known constants. Now, the aforementioned metric has a Gaussian curvature proportional to the equation written before, and the question I'm addressing is, for which $\theta(x,y)$s the generated curvature is zero ?, except for possibly isolated points where it may diverge. Now, the solutions I wrote before correspond to cones, but there should be more analytic solutions.

Any ideas ? Have you seen this equation or someone similar before ?

Thank you so much.

[1] Mostajeran, Cyrus; Warner, Mark; Ware, Taylor H.; White, Timothy J., Encoding Gaussian curvature in glassy and elastomeric liquid crystal solids, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 472, No. 2189, Article ID 20160112, 16 p. (2016). ZBL1371.82141.