Local Petrovskii lacunas close to parabolic singular points of the wavefronts of~strictly hyperbolic partial differential equations

V. A. Vassiliev

Sbornik. Mathematics, Tome 207 (2016) no. 10, pp. 1363-1383

Résumé

We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions of strictly hyperbolic PDEs with constant coefficients in $\mathbb{R}^N$) close to parabolic singular points of their wavefronts (that is, at the points of types $P_8^1$, $P_8^2$, $\pm X_9$, $X_9^1$, $X_9^2$, $J_{10}^1$ and $J_{10}^3$). These points form the next most difficult family of classes in the natural classification of singular points after the so-called simple singularities $A_k$, $D_k$, $E_6$, $E_7$ and $E_8$, which have been investigated previously. Also we present a computer program which counts the topologically distinct morsifications of critical points of smooth functions, and hence also the local components of the complement of a generic wavefront at its singular points. Bibliography: 22 titles.

Détail

Keywords: hyperbolic operator, sharpness, Petrovskii cycle, Petrovskii criterion.
Mots-clés : wavefront, lacuna, morsification