Interior layer at a simple zero of advection ($z=(x-x_{\ast })/\epsilon$)

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.

Reversing the diffusion sign changes which rapid modes remain bounded. In the explicit modelintegration gives $y^{\prime }=C\exp ((x-1{)}^2/{\epsilon }^2)$. ThusThe numerator and denominator are dominated by endpoints. Away from both endpoints, the limit is $(A+B)/2$, since the two endpoint contributions in the denominator are equal. The leading boundary layer factors are $e^{-2x/{\epsilon }^2}$ and $e^{-2(2-x)/{\epsilon }^2}$, of width ${\epsilon }^2$.

On the interior scale $z=(x-1)/\epsilon$, $Y^{\prime \prime }-2z{Y}^{\prime }=0$ has $Y^{\prime }=C{e}^{z^2}$, so a bounded leading inner solution cannot connect unequal constants at both infinities. For equations with extra reaction or forcing, the selected bulk value and smaller interior corrections depend on those terms and their solvability condition; the simple-zero drift alone does not determine them.