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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:


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!


share|cite|improve this question
For $a=0$, $f=0$ (this is just the Laplace equation). – Florian Nov 11 '12 at 9:47
Your system of PDEs is overdetermined. If you know $L$ explicitly, you should be able to explicitly write out the equations you have, and differentiate both sides until you encounter some obstructions, or reach involution (, proving the existence of local solutions. However, since your operator $L$ seems to be mysterious, it would help if you could say more about it. Do you know the order of $L$, or whether $L$ is elliptic or hyperbolic etc., or if $L$ is linear? – Ben McKay Mar 18 '13 at 17:20

The condition on $g$ gives a definite pde for $A$. This can be seen in the following way. Let us insert the solution $f=A(x,y)e^{ig(x,y)}$ into the Helmholtz equation. We get $$ \Delta A+2i(\partial_xg\partial_xA+\partial_yg\partial_yA)+\Phi(x,y)A=0 $$ being $$ \Phi(x,y)=i\Delta g-(\partial_xg)^2-(\partial_yg)^2+a. $$ Now, assuiming $L$ is a linear operator with the Green function $LG=\delta$, one can write $$ g(x,y)=g_0(x,y)+\int_\Omega dx'dy'G(x,x';y,y')h(x',y') $$ being $Lg_0=0$. By substituting this into $\Phi$ and the equation for $A$ we get a partial differential equation to solve. For some operator $L$, the final equation could be simple to manage but, for the general case, maybe some approximation techniques could help.

share|cite|improve this answer
you missed a A before the $\Phi(x,y)$. But yes, thats an idea to directly evaluate the equation. It doesnt give me the actual answere what restrictions there are, but it seems like to be a non-trivial problem. Thanks in any way. I will try some FEMs to solve it, if there is no simple analytical way. – MarkusWave Nov 12 '12 at 0:48

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