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

Abstract

Formal asymptotics are substantiated that describe a typical dropping cusp singularity in the semiclassical approximations to solutions of two cases of the integrable nonlinear Schrödinger 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 the ideas and facts of the mathematical catastrophe theory and the part of Yu.F. Korobeinik’s theorem concerning analytical, as \(h\to 0\), solutions \(G(h,u) \) of the mixed type linear equation \(hG^{\prime \prime }_{hh}=G^{\prime \prime }_{uu}\) to which the hodograph images of both cases of the systems of equations of these semiclassical approximations are equivalent.