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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 :-)

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  • $\begingroup$ It's not quite clear which solution(s) you are interested in. Are you interested in finding just one particular solution or in a class of them? In either case I think you've left the boundary data under-specified. Also, the domain on which you want to study the solution is also not clear. Can you state more precisely the domain you are interested in, in particular excluding regions that are not of physical interest, and also its boundary? $\endgroup$ – Igor Khavkine Oct 15 '14 at 11:55
  • $\begingroup$ @IgorKhavkine Thanks for your comment. I indeed would like to find a particular solution matching the boundary conditions, which are known functions of the angle $\phi$ for two given $z$. I totally agree with you, the problem as I presented is not entirely clear... even for me ! But the domain is clear I think, $x\in\left[-1,1\right]$ and $\phi\in\left]-\pi/2,\pi/2\right[$ for $f$ in $\mathbb{C}$. $\endgroup$ – FraSchelle Oct 15 '14 at 13:46
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If $\epsilon$ is not zero, you need initial conditions in the $\phi$ direction. In general, the resulting solutions will oscillate rapidly when $\epsilon$ is small. Your formal expansion in powers of $\epsilon$ can be expected to approximate a distinguished "slowly varying" solution.

However, although this kind of heuristics is generally deemed good enough for physicists, proving some rigorous version of it is far from easy.

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