Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 009, 21 pages      arXiv:1909.13211      https://doi.org/10.3842/SIGMA.2020.009
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

New Examples of Irreducible Local Diffusion of Hyperbolic PDE's

Victor A. Vassiliev ^ab
a)  Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
^b)  National Research University Higher School of Economics, Moscow, Russia

Received September 29, 2019, in final form February 18, 2020; Published online February 24, 2020

Abstract
Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We radically disprove the obvious conjecture extending this fact to arbitrary singularities: namely, we present examples of diffusion at all non-simple singularity classes of generic wavefronts in odd-dimensional spaces, which are not reducible to diffusion at simple singular points.

Key words: wavefront; discriminant; critical point; morsification; vanishing cycle; hyperbolic PDE; fundamental solution; lacuna; sharp front; diffusion; Petrovskii condition.

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References

1. A'Campo N., Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I, Math. Ann. 213 (1975), 1-32.
2. Arnold V.I., Goryunov V.V., Lyashko O.V., Vassiliev V.A., Singularity theory. II. Applications, Dynamical Systems, Vol. 39, Springer-Verlag, Berlin, 1993.
3. Arnold V.I., Gusein-Zade S.M., Varchenko A.N., Singularities of differentiable maps, Vol. 1, Classification of critical points, caustics, and wavefronts, Birkhäuser, Basel, 2012.
4. Arnold V.I., Gusein-Zade S.M., Varchenko A.N., Singularities of differentiable maps, Vol. 2, Monodromy and asymptotics of integrals, Birkhäuser, Basel, 2012.
5. Atiyah M.F., Bott R., Gårding L., Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math. 124 (1970), 109-189.
6. Atiyah M.F., Bott R., Gårding L., Lacunas for hyperbolic differential operators with constant coefficients. II, Acta Math. 131 (1973), 145-206.
7. Borovikov V.A., Fundamental solutions of linear partial differential equations with constant coefficients, Transact. Moscow Math. Soc. 8 (1959), 199-257, English transl.: Amer. Math. Soc. Transl. Ser. 2, Vol. 25, 1963, 11-76.
8. Davydova A.M., A sufficient condition for the absence of a lacuna for a partial differential equation of hyperbolic type, Ph.D. Thesis, Moscow State University, 1945.
9. Gabrielov A.M., Intersection matrices for certain singularities, Funct. Anal. Appl. 7 (1973), 182-193.
10. Gabrielov A.M., Palamodov V.P., Huygens principle and its generalizations, Comments to the Russian translation of [17], in Petrovskii I., Selected works, Moscow, Nauka, 1986 (in Russian).
11. Gårding L., Sharp fronts of paired oscillatory integrals, Publ. Res. Inst. Math. Sci. 12 (1977), 53-68.
12. Gusein-Zade S.M., Intersection matrices for some singularities of functions of two variables, Funct. Anal. Appl. 8 (1974), 10-13.
13. Leray J., Un prolongementa de la transformation de Laplace qui transforme la solution unitaires d'un opérateur hyperbolique en sa solution élémentaire (Problème de Cauchy. IV), Bull. Soc. Math. France 90 (1962), 39-156.
14. Looijenga E., The complement of the bifurcation variety of a simple singularity, Invent. Math. 23 (1974), 105-116.
15. Looijenga E., The discriminant of a real simple singularity, Compositio Math. 37 (1978), 51-62.
16. Milnor J., Singular points of complex hypersurfaces, Annals of Mathematics Studies, Vol. 61, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1968.
17. Petrowskii I., On the diffusion of waves and the lacunas for hyperbolic equations, Sb. Math. 17(59) (1945), 289-370.
18. Vassiliev V.A., Sharpness and the local Petrovskii condition for strictly hyperbolic equations with constant coefficients, Math. USSR Izv. 28 (1987), 233-273.
19. Vassiliev V.A., Geometry of local lacunae of hyperbolic operators with constant coefficients, Sb. Math. 75 (1993), 111-123.
20. Vassiliev V.A., Applied Picard-Lefschetz theory, Mathematical Surveys and Monographs, Vol. 97, Amer. Math. Soc., Providence, RI, 2002.
21. Vassiliev V.A., Local Petrovskii lacunas close to parabolic singular points of the wavefronts of strictly hyperbolic partial differential equations, Sb. Math. 207 (2016), 1363-1383, arXiv:1607.04042.
22. Vassiliev V.A., A program enumerating real morsifications of isolated real function singularities, available at https://drive.google.com/file/d/1tu4qOfmcsMkQCmwFAStYOYgb-K1R5xG3/view?usp=sharing.
23. Vassiliev V.A., A program enumerating real morsifications of isolated real function singularities of corank $\leq 2$, available at https://drive.google.com/file/d/1RtiqiuBBNYtf218zNRK8SIoFhA6GRb38/view?usp=sharing.