I have a bit of trouble solving a system of first order PDE's, that I get by solving a boundary issue problem in gravitation (here). I have six equations:
$-\frac{2r}{l^2}\zeta^r-\frac{2(r^2+l^2)}{l^2}\partial_t\zeta^t=0$
$-\frac{2rl^2}{(r^2+l^2)^2}\zeta^r+\frac{2l^2}{r^2+l^2}\partial_r\zeta^r=0$
$2r\zeta^r-2r\partial_\phi\zeta^\phi=0$
$-\frac{r^2+l^2}{l^2}\partial_r\zeta^t+\frac{l^2}{r^2+l^2}\partial_t\zeta^t=0$
$-\frac{r^2+l^2}{l^2}\partial_\phi\zeta^t+r^2\partial_t\zeta^\phi=0$
$\frac{l^2}{r^2+l^2}\partial_\phi\zeta^r+r^2\partial_r\zeta^\phi=0$
Where $\zeta^\mu$ ($\mu=t,r,\phi$) is a vector by it's components. And I have the boundary conditions given as subleading terms in $r$ ($\mathcal{O}(r^n)$). Now I've seen that they are supposing the ansatz of the form: $\zeta^\mu=\sum_n \zeta^\mu_n(t,\phi)r^n$. And I've tried to put that in, but I don't really see what would I get. Since I'm expanding the series in Laurent series (n goes to $-\infty$ from $n=0,-1,\ldots, -\infty$), do I have to see what boundary condition I have (for example, for first equation it's $\mathcal{O}(1)$, second it's $\mathcal{O}(r^{-3})$, etc.), and for each equation, I need to expand my solution to that order, ignoring everything beyond that, and try to solve it that way?
I tried putting this in Mathematica, but that doesn't work since I get that the system is overdetermined.
So if you can point me in the right direction, or maybe give me some literature where they show how to solve such systems, I'd really appreciate it.
Thanks
EDIT:
So to add a little background to the story: In gravitation, you have a metric, which basically gives you the means to understand how the time-space behaves. In this certain case, the space in question is called anti de-Sitter space (don't want to go into details, as this is a lengthy subject). You can perturbe this metric by some small amount, but these perturbations must not be too large, otherwise your calculations will go bananas.
But if you give certain boundary conditions, which basically give you 'limit' on how these perturbations can behave, you can extract a lot of interesting stuff out of it, like generators for certain algebras and so on.
So I want to find a diffeomorphism, along this metric, which satisfies the boundary conditions. I do this by making Lie derivative $\mathcal{L}_\zeta g_{\mu\nu}$, for every non zero component of the metric. So by doing that, I get (for the case of AdS$_3$) these 6 partial differential equations, and then by solving them I get $\zeta^\mu=\zeta^t\partial_t+\zeta^r\partial_r+\zeta^\phi\partial_\phi$, $\partial_\mu$ are unit vectors and by them are their components...
I hope this makes it somewhat simpler...
