 Interior layer at a simple zero of advection

A simple zero of an advection coefficient can support an interior layer. For the specific homogeneous equationset $z=(x-x_{\ast })/\epsilon$. The two derivative terms then have the same order, giving $Y^{\prime \prime }+2z{Y}^{\prime }=0$. Hence $Y^{\prime }=C{e}^{-z^2}$ andconnects $Y(-\infty )=A$ and $Y(+\infty )=B$. Its physical width is $O(\epsilon )$. More generally, $a(x_{\ast })=0$, $a^{\prime }(x_{\ast })\ne 0$ in ${\epsilon }^2{y}^{\prime \prime }+a(x)y^{\prime }=0$ gives the same width, while the sign of $a^{\prime }(x_{\ast })$ 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 $Y^{\prime \prime }+a^{\prime }(x_{\ast })z{Y}^{\prime }+c(x_{\ast })Y=f(x_{\ast })$. 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.