Journal of Mathematical Sciences

, Volume 185, Issue 4, pp 638–643 | Cite as

Tilted nonparaxail beams and packets for the wave equation with two spatial variables

  • A. B. PlachenovEmail author

The class of relatively undistorted solutions of the wave equation in the two-dimensional space is constructed. This class includes tilted beam-like and packet-like solutions nonharmonic in time with a Gaussian-type localization. Bibliography: 16 titles.


Spatial Variable Wave Equation 
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  1. 1.
    R. Courant and D. Hilbert, Methods of Mathematical Physics [Russian translation], Gostekhizdat, Moscow (1941).Google Scholar
  2. 2.
    H. Bateman, The Mathematical Analysis of Electrical and Optical Wave Motion on the Basis of Maxwell’s Equations [Russian translation], Nauka, Moscow (1958).Google Scholar
  3. 3.
    P. Hillion, “Generalized phases and nondispersive waves,” Acta Appl. Math., 30, No. 1, 35–45 (1993).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    A. P. Kiselev and M. V. Perel, “Relatively distorsion-free waves for the m-dimensional wave equation,” Diff. Eqs. 38(8), 1206–1207 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    A. P. Kiselev, “Localized light waves: paraxial and exact solutions of the wave equation (review),” Optica Spectroskop., 102(4), 603–622 (2007).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Torre, “Relativistic Laguerre polynomials and splash pulses,” in: Progress in Electromagnetics Research B, 13 (2009), pp. 329–356.Google Scholar
  7. 7.
    A. P. Kiselev and A. B. Plachenov, “Exact solutions of the m-dimensional wave equation form paraxial ones. Further generalizations of the Bateman solution,” Zap. Nauchn. Semin. POMI, 393, 167–177 (2011).MathSciNetGoogle Scholar
  8. 8.
    A. P. Kiselev and M. V. Perel, “Highly localized solutions of the wave equation,” J. Math. Phys., 41(4), 1934–1955 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    J. N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys., 54, 1179–1185 (1983).CrossRefGoogle Scholar
  10. 10.
    A. P. Kiselev, “Modulated Gaussian beams,” Radiophys. Quant. Electron., 26(5), 755–761 (1983).MathSciNetCrossRefGoogle Scholar
  11. 11.
    I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the wave equation,” J. Math. Phys., 30, 1254–1269 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A, 39, 2005–2033 (1989).MathSciNetCrossRefGoogle Scholar
  13. 13.
    M. M. Popov, “Fundamental oscillations of many-mirrow resonators,” Vestn. Leningr. Univ., Ser. Fiz. Khim., 22, No. 4, 42–54 (1969).Google Scholar
  14. 14.
    V. P. Maslov, Complex WKB Method in Nonlinear Equations [in Russian], Nauka, Moscow (1977).Google Scholar
  15. 15.
    Yu. A. Anan’ev, Optical Resonators and Laser Beams [in Russian], Nauka, Moscow (1980).Google Scholar
  16. 16.
    Y. Hadad and T. Melamed, “Parametrization of the tiled Gaussian beam waveobjects,” in: Progress in Electromagnetics Research, 102 (2010), pp. 65–80.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.Moscow State Technical University of Radio Engineering, Electronics and AutomaticsMoscowRussia

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