Solving Ricci flow equation for a $2D$ Kahler manifold

Question

For the $2D$ Kahler manifold, the Ricci flow equation (which is also a one-loop RG equation for the $\sigma$-model on this space) can be written in the form $\frac{\partial^2 \Phi}{\partial u^2}=\frac{\partial \Phi}{\partial u} \frac{\partial \Phi}{\partial \tau}$, where $\Phi$ is related to the conformal factor of the metric, $\Omega(u, \tau)$. The solution gives the behaviour of $\Omega$ as we move along the RG time $\tau$, giving the scale-dependence of the $\sigma$-model QFT. The equation looks simple, so I suspect that it admits an explicit solution, but I can't find it.

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

Comments

To be clear, you are explicitly and only asking for solutions about the PDE $\frac{\partial^2 \Phi}{\partial u^2}=\frac{\partial \Phi}{\partial u} \frac{\partial \Phi}{\partial \tau}$?

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

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@EmilioPisanty Yep.

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

Answer 1

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