Typical dropping asymptotics of quasiclassical approximations to solutions of the nonlinear Schrödinger equation

Formal asymptotics are substantiated that describe typical dropping cusp singularity of quasiclassical approximations to solutions of two cases of the integrable nonlinear Schrödinger equation $-i\epsilon {\mathrm{\Psi }}_t^{\mathrm{\prime }}={\epsilon }^2{\mathrm{\Psi }}_{xx}^{\mathrm{\prime }\mathrm{\prime }}\pm 2\mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^2\mathrm{\Psi }$, where $\epsilon$ 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 $h{G}_{hh}^{\mathrm{\prime }\mathrm{\prime }}=G_{uu}^{\mathrm{\prime }\mathrm{\prime }}$ to which the hodograph images of both cases of the systems of equations of these quasiclassical approximations are equivalent.