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 Mat. Sb., 2016, Volume 207, Number 10, Pages 4–27 (Mi msb8720)

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

V. A. Vassilievab

a National Research University "Higher School of Economics" (HSE), Moscow
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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.

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

 Funding Agency Grant Number Russian Science Foundation 16-11-10316 Research was supported by a Russian Science Foundation grant (project no. 16-11-10316).

DOI: https://doi.org/10.4213/sm8720

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English version:
Sbornik: Mathematics, 2016, 207:10, 1363–1383

Bibliographic databases:

ArXiv: 1607.04042
UDC: 517.955+515.16
MSC: Primary 35L30, 58G17; Secondary 38K40

Citation: V. A. Vassiliev, “Local Petrovskii lacunas close to parabolic singular points of the wavefronts of strictly hyperbolic partial differential equations”, Mat. Sb., 207:10 (2016), 4–27; Sb. Math., 207:10 (2016), 1363–1383

Citation in format AMSBIB
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• https://doi.org/10.4213/sm8720
• http://mi.mathnet.ru/eng/msb/v207/i10/p4

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This publication is cited in the following articles:
1. V. V. Zharinov, “Hamiltonian operators in differential algebras”, Theoret. and Math. Phys., 193:3 (2017), 1725–1736
2. Victor A. Vassiliev, “New Examples of Irreducible Local Diffusion of Hyperbolic PDE's”, SIGMA, 16 (2020), 009, 21 pp.
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