  Formal asymptotics are substantiated that describe typical dropping cusp
singularity of quasiclassical approximations to solutions of two cases of the
integrable nonlinear Schr\"odinger equation
$-i\varepsilon\Psi'_{t}=\varepsilon^2\Psi''_{xx}\pm2|\Psi| ^2\Psi$, where
$\varepsilon$ is a small parameter. The substantiation uses the ideology and
facts of the mathematical catastrophe theory and the part of the theorem of Yu.
F. Korobeinik, concerning analytical as $h\to 0$ solutions $G(h,u)$ of the
mixed type linear equation $hG''_{hh}=G''_{uu}$ to which the hodograph images
of both cases of the systems of equations of these quasiclassical
approximations are equivalent.
