Hörmander has presented a remarkable example of a solution of the homogeneous wave equation, which has a singularity at a running point. An analytic investigation of this solution is performed for the case of three spatial variables. The support of this solution is described, its behavior near the singular point is studied, and its local integrability is established. It is observed that the Hörmander solution is a specialization of a solution found by Bateman five decades in advance.
Similar content being viewed by others
References
L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Springer, Berlin (1983).
H. Bateman, “The conformal transformations of space of four dimensions and their applications to geometrical optics,” Proc. London Math. Soc., No. 7, 70–89 (1909).
H. Bateman, The Mathematical Analysis of Electrical and Optical Wave-Motion on the Basis of Maxwell’s Equations, Dover, New York (1955).
L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Oxford, Pergamon (1971).
P. Hillion, “The Courant–Hilbert solutions of the wave equation,” J. Math. Phys., 33, No. 8, 2749–2753 (1992).
A. P. Kiselev and M. V. Perel, “Highly localized solutions of the wave equation,” J. Math. Phys., 41, No. 4, 1934–1955 (2000).
A. P. Kiselev, “Localized light waves: Paraxial and exact solutions of the wave equation (a review),” Optics and Spectroscopy, 102, No. 4, 603–622 (2007).
I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, “Bateman conformal transformations within the framework of the bidirectional spectral representation,” Progress in Electromagnetics Research, 48, 201–231 (2004).
A. P. Kiselev and A. B. Plachenov, “Exact solutions of the m-dimensional wave equation from paraxial ones. Further generalizations of the Bateman solution,” J. Math. Sci., 185, No. 4, 605–610 (2012).
V. S. Vladimirov, Equations of Mathematical Physics, Mir, Moscow (1985).
A. S. Blagoveshchensky, “Plane waves, Bateman’s solutions, and sources at infinity,” J. Math. Sci., 214, No. 3, 260–267 (2016).
A. S. Blagovestchenskii, A. P. Kiselev, and A. M. Tagirdzhanov, “ Simple solutions of the wave equation with a singularity at a running point, passed on the complexified Bateman solution,” J. Math. Sci., 224, No. 47, 73–82 (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 471, 2018, pp. 76–85.
Rights and permissions
About this article
Cite this article
Blagoveshchensky, A.S., Tagirdzhanov, A.M. & Kiselev, A.P. On the Bateman–Hörmander Solution of the Wave Equation Having a Singularity at a Running Point. J Math Sci 243, 682–688 (2019). https://doi.org/10.1007/s10958-019-04569-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-019-04569-3