= Oscillatory endpoint with a quadratically vanishing drift
{title2=$s_t\sim\sqrt2\epsilon^{1/4},\quad \Delta s=O(\epsilon^{5/12})$}
For $\epsilon y''+s^2y'+y=0$ on $s\geq0$, removing the drift by $y=e^{-s^3/(6\epsilon)}v$ gives
$$
\epsilon^2v''=(s^4/4+\epsilon s-\epsilon)v.
$$
The transformed potential changes sign at $s_t\sim\sqrt2\epsilon^{1/4}$. <WKB approximation> is oscillatory inside and exponential outside this turning region; an <Airy turning-point connection formula> applies over width $\epsilon^{5/12}$. At the endpoint the wavelength is $\sqrt\epsilon$, while the physical exponential prefactor changes on the envelope scale $\epsilon^{1/3}$. Balancing only diffusion and drift and declaring a single $\epsilon^{1/3}$ layer misses the zeroth-order term and the turning region. Matching constants may be singular at homogeneous boundary-value resonances.
Back to article page
