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

Bibliographie

[1] N. A'Campo, “Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I”, Math. Ann., 213:1 (1975), 1–32 | DOI | MR | Zbl

[2] N. A'Campo, “Le groupe de monodromie du déploiement des singularités isolées de courbes planes. II”, Proceedings of the International congress of mathematicians (Vancouver, B. C., 1974), v. 1, Canad. Math. Congress, Montreal, Que., 1975, 395–404 | MR | Zbl

[3] V. I. Arnol'd, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps, v. I, Monogr. Math., 82, The classification of critical points, caustics and wave fronts, Birkhäuser, Boston, MA, 1985, xi+382 pp. | DOI | MR | Zbl | Zbl

[4] V. I. Arnol'd, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps, v. II, Monogr. Math., 83, Monodromy and asymptotics of integrals, Birkhäuser, Boston, MA, 1988, viii+492 pp. | DOI | MR | Zbl | Zbl

[5] V. I. Arnol'd, V. V. Goryunov, O. V. Lyashko, V. A. Vasil'ev, “Singularity theory. II. Classification and applications”, Dynamical systems – VIII, Encyclopaedia Math. Sci., 39, Springer, Berlin, 1993, 1–235 | DOI | MR | Zbl | Zbl

[6] M. F. Atiyah, R. Bott, L. Gårding, “Lacunas for hyperbolic differential operators with constant coefficients. I”, Acta Math., 124 (1970), 109–189 | DOI | MR | Zbl

[7] M. F. Atiyah, R. Bott, L. Gårding, “Lacunas for hyperbolic differential operators with constant coefficients. II”, Acta Math., 131 (1973), 145–206 | DOI | MR | Zbl

[8] V. A. Borovikov, “Fundamental solutions of linear partial differential equations with constant coefficients”, Amer. Math. Soc. Transl. Ser. 2, 25, Amer. Math. Soc., Providence, R.I., 1963, 11–76 | MR | Zbl

[9] V. A. Borovikov, “Nekotorye dostatochnye usloviya otsutstviya lakun”, Matem. sb., 55(97):3 (1961), 237–254 | MR | Zbl

[10] A. M. Davydova, Dostatochnoe uslovie otsutstviya lakuny dlya differentsialnogo uravneniya v chastnykh proizvodnykh giperbolicheskogo tipa, Diss. $\dots$ kand. fiz.-matem. nauk, MGU, M., 1945, 43 pp.

[11] A. M. Gabrièlov, “Intersection matrices for certain singularities”, Funct. Anal. Appl., 7:3 (1973), 182–193 | DOI | MR | Zbl

[12] L. Gårding, “Sharp fronts of paired oscillatory integrals”, Publ. Res. Inst. Math. Sci., 12:99 (1976), 53–68 | DOI | MR | Zbl

[13] S. M. Gusein-Zade, “Intersection matrices for certain singularities of functions of two variables”, Funct. Anal. Appl., 8:1 (1974), 10–13 | DOI | MR | Zbl

[14] S. M. Gusein-Zade, “Dynkin diagrams for singularities of functions of two variables”, Funct. Anal. Appl., 8:4 (1974), 295–300 | DOI | MR | Zbl

[15] S. M. Gusein-Zade, “The monodromy groups of isolated singularities of hypersurfaces”, Russian Math. Surveys, 32:2 (1977), 23–69 | DOI | MR | Zbl

[16] J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Hermann, Paris, 1932, 542 pp. | MR | Zbl

[17] J. Leray, “Un prolongement de la transformation de Laplace qui transforme la solution unitaire d'un opérateur hyperbolique en sa solution élémentaire. (Problème de Cauchy. IV)”, Bull. Soc. Math. France, 90 (1962), 39–156 | MR | MR | Zbl

[18] I. Petrowsky, “On the diffusion of waves and the lacunas for hyperbolic equations”, Matem. sb., 17(59):3 (1945), 289–370 | MR | Zbl

[19] A. N. Varchenko, “On normal forms of nonsmoothness of solutions of hyperbolic equations”, Math. USSR-Izv., 30:3 (1988), 615–628 | DOI | MR | Zbl

[20] V. A. Vasil'ev, “Sharpness and the local Petrovskii condition for strictly hyperbolic operators with constant coefficients”, Math. USSR-Izv., 28:2 (1987), 233–273 | DOI | MR | Zbl

[21] V. A. Vasil'ev, “Geometry of local lacunae of hyperbolic operators with constant coefficients”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 111–123 | DOI | MR | Zbl

[22] V. A. Vassiliev, Applied Picard–Lefschetz theory, Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, 2002, xii+324 pp. | DOI | MR | Zbl