Geodesic
On the Bateman–Hörmander solution of the wave equation, having a singularity at a running point
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 48, Tome 471 (2018), pp. 76-85 
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. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 48, Tome 471 (2018), pp. 76-85. http://geodesic.mathdoc.fr/item/ZNSL_2018_471_a4/
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     title = {On the {Bateman{\textendash}H\"ormander} solution of the wave equation, having a~singularity at a~running point},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] L. Hörmander, The Analysis of Linear Partial Differential Operators, v. I, Distribution Theory and Fourier Analysis, Springer, Berlin, 1983 | MR | Zbl

[2] H. Bateman, “The conformal transformations of space of four dimensions and their applications to geometrical optics”, Proc. London Math. Soc., 7 (1909), 70–89 | DOI | MR | Zbl

[3] H. Bateman, The Mathematical Analysis of Electrical and OpticalWave-Motion on the Basis of Maxwell's Equations, Dover, NY, 1955 | MR

[4] L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, Pergamon, Oxford, 1971 | MR

[5] P. Hillion, “The Courant–Hilbert solutions of the wave equation”, J. Math. Phys., 33:8 (1992), 2749–2753 | DOI | MR | Zbl

[6] A. P. Kiselev, M. V. Perel, “Highly localized solutions of the wave equation”, J. Math. Phys., 41:4 (2000), 1934–1955 | DOI | MR | Zbl

[7] A. P. Kiselev, “Localized light waves: Paraxial and exact solutions of the wave equation (a review)”, Optics and Spectroscopy, 102:4 (2007), 603–622 | DOI

[8] I. M. Besieris, A. M. Shaarawi, A. M. Attiya, “Bateman conformal transformations within the framework of the bidirectional spectral representation”, Progress in Electromagnetics Research, 48 (2004), 201–231 | DOI

[9] A. P. Kiselev, A. B. Plachenov, “Exact solutions of the $m$-dimensional wave equation from paraxial ones. Further generalizations of the Bateman solution”, J. Math. Sci., 185:4 (2012), 605–610 | DOI | MR | Zbl

[10] V. S. Vladimirov, Equations of Mathematical Physics, Mir, Moscow, 1985 | MR

[11] A. S. Blagoveshchensky, “Plane waves, Bateman's solutions, and sources at infinity”, J. Math. Sci., 214:3 (2016), 260–267 | DOI | MR | Zbl

[12] A. S. Blagovestchenskii, A. P. Kiselev, A. M. Tagirdzhanov, “Simple solutions of the wave equation with a singularity at a running point, pased on the complexified Bateman solution”, J. Math. Sci., 224:1 (2017), 47–53 | DOI | MR | Zbl