Umbilic Singularities of Semiclassical Approximations to Solutions of the Focusing Nonlinear Schrödinger Equation

A complete asymptotic expansion of solutions to the elliptic system of nonlinear geometric optics near their singular points typical from the point of view of the mathematical catastrophe theory and related to the singularity of a section of the elliptic umbilic equation is constructed and rigorously substantiated. The substantiation uses known facts of the theory of singularities of smooth mappings and É. Picard’s result concerning analytic solutions of a general nonautonomous linear elliptic equation with two independent variables. The conclusion of the well-known 2009 paper of B. A. Dubrovin, T. Grava, and C. Klein on the normal form used to describe such typical umbilic singularities is improved.