 Large-wavenumber branches of kinetic solidification

The cubic follows by eliminating $\sigma =p^2-Vp-{\alpha }^2$ from one-sided kinetic solidification dispersion, with $V=1-S$. For $\Gamma =0$, the interface branch tends to $\sigma =SV-SV/\alpha +O({\alpha }^{-2})$. For fixed $0<\Gamma <1$, substituting $p=\sqrt{1-\Gamma }\alpha +d$ gives $2d-V=S\Gamma /(1-\Gamma )$, hence $\sigma =-\Gamma {\alpha }^2+[S\Gamma /\sqrt{1-\Gamma }]\alpha +O(1)$. At $\Gamma =1$ the balance is $p^3\sim S{\alpha }^2$, giving $\sigma =-{\alpha }^2+S^{2/3}{\alpha }^{4/3}+o({\alpha }^{4/3})$. For $1<\Gamma <1+2S/V$, $p\to S/(\Gamma -1)>V/2$ and $\sigma =-{\alpha }^2+p^2-Vp+o(1)$. Stronger capillarity can remove the real isolated interface mode from the chosen thermal branch; it does not create growth. These distinctions explain why replacing $p$ by $\alpha$ is not a controlled fixed-capillarity limit.