Guide Propagation and Interaction of Plasma Waves. Metamaterials

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


In this chapter we sketch the basic mathematical equations, adding general relations and inhomogeneity or boundary conditions as a reason for guide formation in a plasma, and illustrating them with simple examples.


  1. 1.
    V.I. Petviahsvili, Fiz. Plasmy SSSR I, 28 (1975)Google Scholar
  2. 2.
    V.L. Ginsburg, A.A. Rukhadze, Waves in Magnetoactive Plasma (Nauka, Moscow, 1975)Google Scholar
  3. 3.
    V.P. Silin, Introduction to the Kinetic Theory of Gases (Lebedev Inst Press, Moscow, 1998), in RussianGoogle Scholar
  4. 4.
    A.A. Vlasov, J. Exp. Theor. Phys. 8, 291 (1938)Google Scholar
  5. 5.
    L.D. Landau, J. Exp. Theor. Phys. 16, 574 (1946)Google Scholar
  6. 6.
    S.B. Leble, Waveguide Propagation of Nonlinear Waves in Stratified Media (Leningrad University Press, Leningrad, 1988), in Russian; extended edn. (Springer, Berlin, 1990)Google Scholar
  7. 7.
    D. Bohm, E.P. Gross, Theory of plasma oscillations. Excitation and damping of oscillations. Phys. Rev. 75, 1864 (1949)ADSCrossRefGoogle Scholar
  8. 8.
    V.P. Silin, V.T. Tikhonchuk, J. Exp. Theor. Phys. 81, 2039–2051 (1981)Google Scholar
  9. 9.
    V.M. Babich, V.S. Buldyrev, I.A. Molotkov, Space-Time Ray Method: Linear and Nonlinear Waves (Leningrad University Press, Leningrad, 1985)Google Scholar
  10. 10.
    I.V. Karpov, S.B. Leble, V.M. Smertin, Geomagnet. Aeron. SSSR 4, 672–673 (1983)ADSGoogle Scholar
  11. 11.
    A.A. Zaitsev, S.B. Leble, Theory of Nonlinear Waves (Kaliningrad University Press, Kaliningrad, 1984)Google Scholar
  12. 12.
    V.E. Zakharov, S.V. Manakov, S.P. Novikov, J.P. Pitaevski, Theory of Solitons. The Method of Inverse Problems (Nauka, Moscow 1980); [English: Plenum, New York 1984]Google Scholar
  13. 13.
    V.I. Petviashvili, Vopr. Teor. Piaz. 9(11), 59–82 (1979)Google Scholar
  14. 14.
    L.M. Gorbunov, V.P. Silin, J. Exp. Theor. Phys. 47, 203–210 (1964)Google Scholar
  15. 15.
    L.M. Gorbunov, A.M. Tunerbulatov, J. Exp. Theor. Phys. 53, 1494–1497 (1967)Google Scholar
  16. 16.
    V.P. Maslov: Mathematical Aspects of Integral Optics, Moscow Institute of Electronic Engineering (1983)Google Scholar
  17. 17.
    SYu. Dobrokhotov, V.P. Maslov, Soviet Science Review, vol. 3 (Overseas Publishing Association, Harwood, 1982), pp. 221–311Google Scholar
  18. 18.
    V.E. Zakharov, J. Exp. Theor. Phys. 62, 1745–1755 (1972)Google Scholar
  19. 19.
    V.I. Talanov, ZhETF Pis. Red. 2, 223 (1965) [JETP Lett. 2, 141 (1965)]; V.E. Zakharov: J. Appl. Mech. Tech. Phys. 9, 190 (1968)Google Scholar
  20. 20.
    V.E. Zakharov, A.B. Shabat: (1971) Exact theory of two-dimensional self- focusing and one-dimensional modulation of waves in nonlinear media. Zhurn. Eksp. Teor. Fiz. 61, 118–134 [(1972). Sov. Phys. JETP 34, 62–69]Google Scholar
  21. 21.
    S.B. Leble, D.W. Rohraff, Nonlinear evolution of components of an electromagnetic field of helicoidal waves in plasma. Phys. Scr. 123, 140–144 (2006)CrossRefGoogle Scholar
  22. 22.
    V.P. Dmitriyev, Helical waves on a vortex filament. Am. J. Phys. 73, 563 (2005). Scholar
  23. 23.
    G. Sato, W. Oohara, R. Hatakeyama, Plasma production by helicon waves with single mode number in low magnetic fields, in 12th International Congress on Plasma Physics, 25–29 October 2004, Nice (France)Google Scholar
  24. 24.
    E. Doktorov, S.B. Leble, Dressing Method in Mathematical Physics (Springer, Berlin, 2007)Google Scholar
  25. 25.
    B.G. Konopelchenko, Introduction to Multidimensional Integrable Equations: The Inverse Spectral inverse spectral transform in 2+1 dimensions (Plenum Press, New York, 1992)Google Scholar
  26. 26.
    S. Shinohara, K. Shamrai, Direct comparison of experimental and theoretical results on the antenna loading and density jumps in a high pressure helicon source. Plasma Phys. Control. Fusion 42, 865–880 (2000)ADSCrossRefGoogle Scholar
  27. 27.
    B.W. Maxfield, Helicon waves in solids. Am. J. Phys. 37(3), 241–269 (1969)ADSCrossRefGoogle Scholar
  28. 28.
    S. Leble, M. Salle, The Darboux transformations for the discrete analogue of the Silin-Tikhonchuk equation. Dokl. AN SSSR 284, 110–114 (1985)MathSciNetGoogle Scholar
  29. 29.
    T.J. Cui, D.R. Smith, R. Liu (eds): Metamaterials: Theory, Design, and Applications (Springer, Berlin, 2010)Google Scholar
  30. 30.
    R.W. Ziolkowski, A. Kipple, Causality and double-negative metamaterials. Phys. Rev. E 68, 026615 (2003)ADSCrossRefGoogle Scholar
  31. 31.
    R.W. Ziolkowski, F. Auzanneau, Passive artificial molecule realizations of dielectric materials. J. Appl. Phys. 82, 3195–3198 (1997)ADSCrossRefGoogle Scholar
  32. 32.
    K.V. Pravdin, I.Y. Popov, Layered system with metamaterials. J. Phys.: Conf. Ser. 661(1), 012025 (2015)Google Scholar
  33. 33.
    A.A. Perelomova, Projectors in nonlinear evolution problem: acoustic solitons of bubbly liquid. Appl. Math. Lett. 13, 93–98 (2000); Nonlinear dynamics of vertically propagating acoustic waves in a stratified atmosphere. Acta Acustica 84 (6), 1002–1006 (1998)MathSciNetCrossRefGoogle Scholar
  34. 34.
    S.V. Sazonov, N.V. Ustinov, New class of extremely short electromagnetic solitons. General class of the traveling waves propagating in a near oppositely-directional coupler. Pis’ma v Zh. Eksper. Teoret. Fiz. 83(11), 573–578 (2006)ADSCrossRefGoogle Scholar
  35. 35.
    T. Schäfer, C.E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media. Phys. D 196, 90–105 (2004)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Y. Chung, C.K.R.T. Jones, T. Schäfer, C.E. Wayne, Ultra-short pulses in linear and nonlinear media. Nonlinearity 18, 1351–1374 (2004)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Z. Zhaqilao, Q. Hu, Z. Qiao, Multi-soliton solutions and the Cauchy problem for a two-component short pulse system. Nonlinearity 30(10), 3773 (2017)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    P. Kinsler, Phys. Rev. A 81, 023808 (2010)ADSCrossRefGoogle Scholar
  39. 39.
    V.V. Belov, SYu. Dobrokhotov, T.Y.A. Tudorovskiy, Operator separation of variables for adiabatic problem in quantum and wave mechanics. J. Eng. Math. 55(1–4), 183–237 (2006)MathSciNetCrossRefGoogle Scholar
  40. 40.
    M. Kuszner, S. Leble, Directed electromagnetic pulse dynamics: projecting operators method. J. Phys. Soc. Jpn. 80, 024002 (2011)ADSCrossRefGoogle Scholar
  41. 41.
    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)ADSCrossRefGoogle Scholar
  42. 42.
    M. Kuszner, S. Leble, Ultrashort opposite directed pulses dynamics with Kerr effect and polarization account. J. Phys. Soc. Jpn. 83, 034005 (2014)ADSCrossRefGoogle Scholar
  43. 43.
    K. Porsezian, V.C. Kuriakose, Optical Solitons (Springer, Berlin, 2003)CrossRefGoogle Scholar
  44. 44.
    S. Pitois, G. Millot, S. Wabnitz, Nonlinear polarization dynamics of counterpropagating waves in an isotropic optical fiber: theory and experiments. J. Opt. Soc. Am. B 18(4) (2001)ADSCrossRefGoogle Scholar
  45. 45.
    S. Leble, D. Ampilogov, Directed electromagnetic wave propagation in 1D metamaterial: projecting operators method phys. Lett. A 380(29–30), 2271–2278 (2016)ADSMathSciNetGoogle Scholar
  46. 46.
    D. Ampilogov, S. Leble, General equation for directed electromagnetic wave propagation in 1D metamaterial: projecting operator method. TASK Q. 20(2) (2016)Google Scholar
  47. 47.
    D. Ampilogov, S. Leble, Interaction of orthogonal-polarized waves in 1D metamaterial with Kerr nonlinearity, arXiv:1802.09523 [physics.optics]; D. Ampilogov: Interaction of orthogonal-polarized waves in 1D-metamaterial. TASK Q. 21(2), 605–619 (2017)
  48. 48.
    S. Leble, A. Perelomova, Dynamical Projectors Method in Hydro- and Electrodynamics (CRC Press, Taylor and Francis, Boca Raton, 2018)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Immanuel Kant Baltic Federal UniversityKaliningradRussia

Personalised recommendations