= Interior layer at a simple zero of advection
{title2=$z=(x-x_*)/\varepsilon$}
A simple zero of an advection coefficient can support an <interior layer>. For the specific homogeneous equation
$$
\varepsilon^2y''+2(x-x_*)y'=0,
$$
set $z=(x-x_*)/\varepsilon$. The two <derivative> terms then have the same order, giving $Y''+2zY'=0$. Hence $Y'=C e^{-z^2}$ and
$$
Y(z)=\frac{A+B}{2}+\frac{B-A}{2}\operatorname{erf}z
$$
connects $Y(-\infty)=A$ and $Y(+\infty)=B$. Its physical width is $O(\varepsilon)$. More generally, $a(x_*)=0$, $a'(x_*)\ne0$ in $\varepsilon^2y''+a(x)y'=0$ gives the same width, while the sign of $a'(x_*)$ determines whether the nonconstant Gaussian derivative decays.
An order-one reaction term cannot be hidden in this balance: adding $c(x)y=f(x)$ gives the leading inner equation $Y''+a'(x_*)zY'+c(x_*)Y=f(x_*)$. An arbitrary pair of constants need not match that equation. The <error function> connection above requires the extra terms to vanish or be negligible at leading order; reversing diffusion also changes the bounded matching structure.
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