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Evolution Operator and Projectors to Its Eigenspaces

  • Sergey LebleEmail author
Chapter
Part of the Springer Series on Atomic, Optical, and Plasma Physics book series (SSAOPP, volume 109)

Abstract

In this chapter we sketch the basic mathematical notions of dynamical projection  used in this book, starting from general relations and illustrating them by the simplest examples, following [1, 2, 3] and paying particular attention to the impact of inhomogeneities  and accompanying effects. As mentioned in the introduction [see ( 1.1)], in the waveguide propagation, after expanding all the fields in series over the transverse coordinate basis, the coefficients \(\psi _k\) of the expansions will depend on the unique longitudinal space coordinate, say x, and time. Let \(\partial ={\partial }/{\partial x}\) denote the space derivative.

References

  1. 1.
    S.B. Leble, A.A. Zaitsev, Novye Methody v Teorii Nelinejnych Voln [New Methods in Nonlinear Wave Theory] (in Russian) (Kaliningrad University Press, 1987)Google Scholar
  2. 2.
    S. Leble, General remarks on dynamic projection method. TASK Q. 20(2), 113–130 (2016)Google Scholar
  3. 3.
    S. Leble, A. Perelomova, The Dynamical Projector Method: Hydro and Electrodynamics (Taylor and Francis, New York, 2018)Google Scholar
  4. 4.
    A. Perelomova, Development of linear projecting in studies of non-linear flow. Acoustic heating induced by non-periodic sound. Phys. Lett. A 357, 42–47 (2006)CrossRefADSGoogle Scholar
  5. 5.
    A. Perelomova, Driving force of acoustic streaming caused by aperiodic sound beam in unbounded volumes. Ultrasonics 49, 583–587 (2009)CrossRefGoogle Scholar
  6. 6.
    A. Perelomova, Interaction of acoustic and thermal modes in the gas with nonequilibrium chemical reactions. Possibilities Acoust. Cool. Acta Acust. United Acust. 96, 43–48 (2010)CrossRefGoogle Scholar
  7. 7.
    A. Perelomova, P. Wojda, Generation of the vorticity motion by sound in a chemically reacting gas. Invers. Acoust. Streaming Non-Equilib. Regime, Cent. Eur. J. Phys. 9(3), 740–750 (2011)Google Scholar
  8. 8.
    A. Perelomova, P. Wojda, Generation of the vorticity mode by sound in a vibrationally relaxing gas. Cent. Eur. J. Phys. 10(5), 1116–1124 (2012)Google Scholar
  9. 9.
    A. Perelomova, W. Pelc-Garska, Efficiency of acoustic heating produced in the thermoviscous flow of a fluid with relaxation. Cent. Eur. J. Phys. 8(6), 855–863 (2010)Google Scholar
  10. 10.
    A. Perelomova, Interaction of acoustic and thermal modes in the vibrationally relaxing gases. Acoust. Cool. Acta Phys. Pol. A 123(4), 681–687 (2013)CrossRefGoogle Scholar
  11. 11.
    S. Leble, S. Vereshchagin, A wave diagnostics in geophysics: algorithmic extraction of atmosphere disturbance modes. Pure Appl. Geophys. 175(8), 3023–3035 (2018)CrossRefADSGoogle Scholar
  12. 12.
    S. Leble, A. Perelomova, Problem of proper decomposition and initialization of acoustic and entropy modes in a gas affected by the mass force. Appl. Math. Model. 37(3), 629–635 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Perelomova, Nonlinear dynamics of vertically propagating acoustic waves in a stratified atmosphere. Acta Acust. 84(6), 1002–1006 (1998)Google Scholar
  14. 14.
    A. Perelomova, Nonlinear dynamics of directed accoustic waves in stratified and inhomogeneous liquids and gases with arbitrary equation of state. Arch. Acoust. 25, 451–463 (2000)Google Scholar
  15. 15.
    S. Leble, I. Vereshchagina, The method of dynamic projection operators in the theory of hyperbolic systems of partial differential equations with variable coefficients. arXiv:1403.7751
  16. 16.
    S.B. Leble, Waveguide Propagation of Nonlinear Waves in Stratified Media (in Russian), Leningrad University Press, 1988. Extended edn. in (Springer, Berlin 1990)Google Scholar
  17. 17.
    V.V. Belov, S.Y. Dobrohotov, T.Y. Tudorovskiy, Operator separation of variables for adiabatic problems in quantum and wave mechanics. J. Eng. Math. 55(1–4), 183–237 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Immanuel Kant Baltic Federal UniversityKaliningradRussia

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