= Collar-to-film capillary matching
{title2=$H^3H'''=H-1$}
Near a collar edge adjoining a film of thickness $h_f$, local <mass conservation> gives $h^3h'''=c(h-h_f)$. The scaling $h=h_fH$, $x-x_f=h_fX/c^{1/3}$ yields the displayed equation. Its linearization at $H=1$ has exponents $1$ and $-1/2\pm i\sqrt3/2$. The positive growing rear branch approaching one as $X\to-\infty$ has one amplitude, removable by translation; the front's decaying oscillatory branches have two amplitudes, leaving one parameter after translation. This dimension count describes local branches; global positivity and selection require matching. If their core asymptotes, after removing linear terms by translation, are $H_-\sim a_-X^2+b_-$ and $H_+\sim a_+(\lambda)X^2+b_+(\lambda)$, matching to the collar gives $A/2=a_\pm c^{2/3}/h_\pm$ and $B=b_\pm h_\pm$. Consequently $a_+b_+=a_-b_-$; a selected root with $\alpha=a_+(\lambda_0)>a_-$ yields $c=(Ah_+/(2\alpha))^{3/2}$, $B=h_+b_+(\lambda_0)$ and $h_-/h_+=a_-/\alpha$. Local linearization alone does not establish uniqueness of this global matching root.
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