Local Petrovskii lacunas at parabolic singular points of wavefronts of strictly hyperbolic PDE's

TL;DR

This paper classifies local Petrovskii lacunas at parabolic singular points of wavefronts in strictly hyperbolic PDEs, advancing understanding of solution regularity near complex singularities.

Contribution

It introduces a classification of Petrovskii lacunas at parabolic singular points and develops a computer program for counting topologically distinct morsifications.

Findings

Classified Petrovskii lacunas at parabolic singular points.

Developed a computational tool for topological counting of morsifications.

Enhanced understanding of wavefront singularities in hyperbolic PDEs.

Abstract

We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions) of strictly hyperbolic PDE's with constant coefficients in $R^N$ at the parabolic singular points of their wavefronts. These points form the next difficult family of classes of the natural classification of singular points after the so-called simple singularities, studied previously. Also we promote a computer program counting for topologically different morsifications of critical points of smooth functions, and hence also for local components of the complement of a generic wavefront at its singular points.