Description of the PDE $u_{\xi\xi}=a^2u_{yy}-u_{\xi}$

Question

I have the following PDE:

$$\frac{\partial^2u}{\partial\xi^2}=a^2\frac{\partial^2u}{\partial y^2}-\frac{\partial u}{\partial\xi}$$

First of all, this is a PDE, right? And after that is nonlinear and is it homogeneous?