= One-third-power scaling at a nonlinear turning endpoint
{title2=$x=\epsilon^{1/3}X,\quad y=\epsilon^{1/3}Y$}
In $\epsilon y''+xy+y^2=0$, balancing all three terms where the outer branches $0$ and $-x$ meet gives the displayed <dominant balance> and the inner equation $Y''+XY+Y^2=0$. Matching to $-X$ and writing $Y=-X+u$ gives $u''-Xu+u^2=0$. Its linearized decaying correction is an <Airy function>; the growing correction is excluded. Matching to zero instead gives $Y''+XY=0$, whose nonzero <Airy function> combinations oscillate. Thus the acceptable matching branches differ qualitatively at the turning endpoint.
Back to article page
