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Zap. Nauchn. Sem. POMI, 2011, Volume 393, Pages 167–177 (Mi znsl4622)  

This article is cited in 8 scientific papers (total in 8 papers)

Exact solutions of the $m$-dimensional wave equation from paraxial ones. Further generalization of the Bateman solution

A. P. Kiseleva, A. B. Plachenovb

a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia
b Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University), Moscow, Russia

Abstract: A review of earlier generalizations of the classical Bateman solution, involving an arbitrary function, is presented. Its further generalization, described by $m(m-1)$ real parameters characterizing the phase, is given. Under a proper choice of the arbitrary function, it may describe Gaussian beam or Gaussian packet.

Key words and phrases: wave equation, exact solutijns, localized waves, Bateman solutions.

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English version:
Journal of Mathematical Sciences (New York), 2012, 185:4, 605–610

Bibliographic databases:

UDC: 517.95
Received: 08.10.2011

Citation: A. P. Kiselev, A. B. Plachenov, “Exact solutions of the $m$-dimensional wave equation from paraxial ones. Further generalization of the Bateman solution”, Mathematical problems in the theory of wave propagation. Part 41, Zap. Nauchn. Sem. POMI, 393, POMI, St. Petersburg, 2011, 167–177; J. Math. Sci. (N. Y.), 185:4 (2012), 605–610

Citation in format AMSBIB
\Bibitem{KisPla11}
\by A.~P.~Kiselev, A.~B.~Plachenov
\paper Exact solutions of the $m$-dimensional wave equation from paraxial ones. Further generalization of the Bateman solution
\inbook Mathematical problems in the theory of wave propagation. Part~41
\serial Zap. Nauchn. Sem. POMI
\yr 2011
\vol 393
\pages 167--177
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl4622}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2870211}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2012
\vol 185
\issue 4
\pages 605--610
\crossref{https://doi.org/10.1007/s10958-012-0944-7}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866540795}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. B. Plachenov, “Tilted nonparaxial beams and packets for the wave equation with two spatial variables”, J. Math. Sci. (N. Y.), 185:4 (2012), 638–643  mathnet  crossref  mathscinet
    2. M. V. Neschadim, “Klassy obobschennykh funktsionalno invariantnykh reshenii volnovogo uravneniya. I”, Sib. elektron. matem. izv., 10 (2013), 418–435  mathnet
    3. Plachenov A.B., “Obscheastigmaticheskie sosredotochennye resheniya uravneniya Kleina-Fka-Gordona”, Vestnik MGTU MIREA, 2013, no. 1, 137–140  elib
    4. M. V. Neshchadim, “Sphere Generalized Functional Invariant Solutions of Wave Equation”, J. Math. Sci., 211:6 (2015), 805–810  mathnet  crossref
    5. Fialkovsky I.V., Perel M.V., Plachenov A.B., “on Astigmatic Exponentially Localized Solutions For the Wave and the Klein-Gordon-Fock Equations”, J. Math. Phys., 55:11 (2014), 112902  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. S. Blagovestchenskii, A. P. Kiselev, A. M. Tagirdzhanov, “Simple solutions of the wave equation, singular at a ranning point, based on the complexified Bateman solution”, J. Math. Sci. (N. Y.), 224:1 (2017), 47–53  mathnet  crossref  mathscinet
    7. Plachenov A.B., So I.A., Kiselev A.P., “Paraxial Gaussian Modes With Simple Astigmatic Phases and Nonpolynomial Amplitudes”, Proceedings of the International Conference Days on Diffraction (Dd) 2017, eds. Motygin O., Kiselev A., Goray L., Suslina T., Kazakov A., Kirpichnikova A., IEEE, 2017, 264–269  crossref  isi
    8. 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”, Matematicheskie voprosy teorii rasprostraneniya voln. 48, Posvyaschaetsya pamyati Aleksandra Pavlovicha KAChALOVA, Zap. nauchn. sem. POMI, 471, POMI, SPb., 2018, 76–85  mathnet
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