First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads $$\left[\left(\cos\phi\partial_{z}+\mathbf{i}\omega\right)^{2}-\epsilon^{2}\partial_{\phi}^{2}+1\right]f\left(z,\phi\right)=0$$
with the angle $\phi\in\left[0,2\pi\right]$. This equation is hyperbolic unless $\phi = \pm \pi/2$ where it is parabolic, but these points are outside the range of the physics I'm trying to explore, so I can suppose the equation to be hyperbolic I guess. The boundary conditions look like $f\left(\pm1,\phi\right)=f_{R,L}\left(\phi\right)$ and I guess the problem is well posed (since it comes from a physical problem). I'm trying to obtain compact solutions for this equation, but I think it's not possible. The big problem is the $\phi$-dependency of the space-derivative $\partial_{z}$, which makes the equation a wave-like equation with a variable-dependent coefficient (alongside the fact that $\phi$ is an angle and so it has compact range).
So I've tried to obtain some solutions for $\epsilon\ll 1$. I can find some non-trivial solutions using the perturbative Ansatz $f\approx f^{\left(0\right)}\left(x;\phi\right)+\epsilon f^{\left(1\right)}\left(x,\phi\right)+\epsilon^{2}f^{\left(2\right)}\left(x,\phi\right)+\cdots$.
Nevertheless, I'm not at ease with these solutions, since clearly the variable $\phi$ is somehow treated as a parameter, not as a variable in $f^{\left(0\right)}\left(x;\phi\right)$. An other point in favour of something strange when $\epsilon \rightarrow 0$ is that the characteristics of the equation are
$$\xi_{\pm}=x\pm\dfrac{\sin\phi}{\epsilon}$$
and so the limit $\epsilon \rightarrow 0$ is not well defined. NB: One has to cut the space in $\phi\in\left]-\pi/2,\pi/2\right[$ and $\phi\in\left]\pi/2,3\pi/2\right[$ in order to obtain these characteristics, but this is not a big problem I guess, since I can look for solutions in these two interval independently.
So my question is as follow: do you think the equation needs some singular perturbation treatment in the limit $\epsilon \rightarrow 0$ ? If so, where may I find some good references on how to treat such a problem ? I've tried to look at the book by Kevorkian and Cole (Multiple scale and singular perturbation methods - Springer - 1996) but the case when only one differential operator (like in my case, when only $\partial_{\phi}$) is singular is not well documented there. So any suggestion is warm welcome :-)
