Hello! Let's say I have

$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$

with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$).

Now separate the Amplitude and Phase of the solution:

$$f(x,y)=A(x,y)\cdot \exp\big(i\cdot g(x,y)\big)$$

with $A(x,y),g(x,y) \in \mathbb{R}$.

The additional restriction to the Phase g(x,y) is a PDE in the form of:

$\hat{L}g(x,y)=h(x,y)$

Now my questions are:

- Are there restrictions on $h(x,y)$ and differential operator $\hat{L}$ to get a solution $f(x,y)$?
- How can find such a solution f(x,y)? Are there analytical ways? Are there stochastic ways? Are there numerical ways?
- Are there ways to find solutions for the simplification $a=0$?

I'm very thankful for any hint in a useful direction. Unfortunaly I'm totally stuck with this problem. Thanks alot in Advance!

Markus