I have a relatively straightforward differential equation. Actually, the paraxial wave-equation:
\begin{eqnarray} \left(\nabla^{2}-2ik\frac{\partial}{\partial z}\right)\psi=0 \end{eqnarray}
Typically this is solved by choosing a particular coordinate system, for example Cartesian:
\begin{eqnarray} \nabla^{2}&=&\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial x^2},\\ \psi&=&\psi(x,y). \end{eqnarray}
As well as a separable solution: $\psi(x,y)=X(x)Y(y)$.
This is all fine and leads to a lot of interesting stuff, but I want to get "weird". That is,
a.) How do I go about solving the differential equation if I don't want to assume separability (in whatever coordinate system), is there a general non-separable test solution or method that is standard? My PDE knowledge is pretty weak, and everything I was able to find online after a few minutes of searching seemed to deal with PDEs that are non-separable from the start. I'm happy to just be pointed in the right direction if someone has a good online reference handy.
b.) Are there 2-D coordinate systems for which these solutions are well-studied besides Cartesian, cylindrical, and elliptical?