= Solution
At $a=-x_T$, define
$$
\alpha=-\left.\frac{dk^2}{dx}\right|_a=\frac{2\omega_{pe0}^2x_T}{c^2L^2}>0,\qquad
\ell=\alpha^{-1/3},\qquad z=\frac{x-a}{\ell}.
$$
The first-order <Taylor expansion> is $k^2(x)=-\alpha(x-a)+O((x-a)^2)$, so the local <wave equation> becomes
$$
\boxed{\frac{d^2 E}{dz^2}-zE=0,\qquad E=C_A\operatorname{Ai}(z)+C_B\operatorname{Bi}(z).}
$$
This is the <Airy ordinary differential equation>. Its $z<0$ side is oscillatory and its $z>0$ side is evanescent. A matching overlap requires $\ell\ll|x-a|\ll x_T$, hence well-separated <wave turning points> with $\ell\ll x_T$. At $b=x_T$, the coordinate $z_R=(b-x)/\ell$ gives the same <Airy ordinary differential equation>, with its evanescent side again at positive argument.
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