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 Zap. Nauchn. Sem. POMI, 2018, Volume 471, Pages 76–85 (Mi znsl6625)

On the Bateman–Hörmander solution of the wave equation, having a singularity at a running point

A. S. Blagoveshchenskya, A. M. Tagirdzhanovab, A. P. Kiselevcd

a St. Petersburg State University, St. Petersburg, Russia
b St. Petersburg Electrotechnical University, St. Petersburg, Russia
c Steklov Mathematical Institute, St. Petersburg Branch, St. Petersburg, Russia
d Institute of Mechanical Engineering RAS, St. Petersburg, Russia

Abstract: Hörmander have presented a remarkable example of a solution of the homogeneous wave equation, which has a singularity at a running point. We are concerned with analytic investigation of this solution for the case of three spatial variables. We describe its support, study its behavior near the singular point and establish its local integrability. We observe that the Hörmander solution is a specialization of a solution found by Bateman five decades in advance.

Key words and phrases: wave equation, explicit solutions, solutions with a singularity at a running point, Bateman solution, Hörmander solution.

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-00529 This work was supported in part by Russian Foundation for Basic Research grant No. 17-01-00529.

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Document Type: Article
UDC: 517
Language: English

Citation: A. S. Blagoveshchensky, A. M. Tagirdzhanov, A. P. Kiselev, “On the Bateman–Hörmander solution of the wave equation, having a singularity at a running point”, Mathematical problems in the theory of wave propagation. Part 48, Zap. Nauchn. Sem. POMI, 471, POMI, St. Petersburg, 2018, 76–85

Citation in format AMSBIB
\Bibitem{BlaTagKis18} \by A.~S.~Blagoveshchensky, A.~M.~Tagirdzhanov, A.~P.~Kiselev \paper On the Bateman--H\"ormander solution of the wave equation, having a~singularity at a~running point \inbook Mathematical problems in the theory of wave propagation. Part~48 \serial Zap. Nauchn. Sem. POMI \yr 2018 \vol 471 \pages 76--85 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6625}