How to Solve This Coupled PDE System Involving Complex Variables?

[DukeLuke]

$$(r^2 \nabla^2 - 1) X(r,\theta,z) + 2 \frac{\partial}{\partial \theta} Y(r,\theta,z) = 0$$
$$(r^2 \nabla^2 - 1) Y(r,\theta,z) - 2 \frac{\partial}{\partial \theta} X(r,\theta,z) = 0$$


any suggestions are greatly appreciated :)

 

[Ben Niehoff]

I would start with defining

$$Z(r,\theta,z) \equiv X(r,\theta,z) + i Y(r,\theta,z)$$

Then you have less work to do. It looks like separation of variables might be successful, have you tried that? Or a Fourier transform would certainly work.

 

[DukeLuke]

I tried separation of variables, but it didn't work because of the coupling. Using z=x+iy I was able to reduce the system to a single equation (multiply the 2nd equation by i and add it to the first).

$$(r^2\nabla^2-1)Z(r,\theta,z) - 2i \frac{\partial Z(r,\theta,z)}{\partial \theta} = 0$$

Is it necessary that the real and complex parts individually equal 0? I was thinking of trying separation of variables for this equation in Z, but I'm unsure how to deal with the complex portion.