Typical Dropping Asymptotics in the Semiclassical Approximations to Solutions of the Nonlinear Schrödinger Equation

Formal asymptotics are substantiated that describe a typical dropping cusp singularity inthe semiclassical approximations to solutions of two cases of the integrable nonlinearSchrödinger equation \(-i\varepsilon \Psi ^{\prime }_{t} = \varepsilon ^2\Psi^{\prime \prime }_{xx}\pm 2|\Psi | ^2\Psi \) ,where \(\varepsilon\) is a small parameter. The substantiation uses theideas and facts of the mathematical catastrophe theory and the part of Yu.F. Korobeinik’stheorem concerning analytical, as \(h\to 0\) , solutions \(G(h,u)\) of the mixed type linear equation \(hG^{\prime\prime }_{hh}=G^{\prime \prime }_{uu}\) towhich the hodograph images of both cases of the systems of equations of these semiclassicalapproximations are equivalent.