I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I could simplify one of my geometric problems (in the smooth scenario) into the solutions of a system of linear PDEs that is
$$ e_v = L_u\,g,\quad\quad\quad g_u = L_v\,e $$
where $L = \ln(\tan{(\frac{\omega}{2})})$ and $e(u,v),g(u,v),\omega(u,v): [0,a]\times [0,b] \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ are smooth functions on $(0,a)\times (0,b)$. Furthermore, I have the functions $e(u,0)$ and $g(0,v)$ and $\omega(u,v)$ as well. My goal is to show that the system has a solution for $e(u,v)$ and $g(u,v)$. Unfortunately, my knowledge of PDEs is very limited but just from observing the counterpart of the above system in discrete and semi-discrete scenarios I have this kind of feeling that there should be a solution for the above system (with the boundaries that I mentioned).
Can someone clarify how I can prove that there exists a unique solution to the above system. I would very much appreciate if you can also go to details, describing non-characteristic curves and explaining why $e(u,0)$ and $g(0,v)$ are actually enough as boundary conditions (if they are). I think I also have to assume $e(u,v)$ and $g(u,v)$ are analytic functions. Is there any way to avoid this?
Please let me know if you need more details so that I can edit the question.