Actually, it seems that this equation has an analytical solution only for a few initial metrics, which are presented, say, in this paper.

The analytical solution exists, if the boundary conditions allow the solution of the form $\omega (u, \tau)=a(\tau)+b(\tau) \psi(u)$ for $\psi(u)=\mathrm{exp} \left[ \pm \lambda u \right]$, $\mathrm{cosh} \left[\lambda u+A \right]$, $\mathrm{sinh} \left[\lambda u+A \right]$, $\mathrm{cos} \left[\lambda u+A \right]$, and $\frac{1}{\omega}=\frac{\partial \Phi}{\partial u}$.

This post imported from StackExchange Physics at 2017-02-16 08:55 (UTC), posted by SE-user Andrey Feldman