Plasma Kinetic Theory: Vlasov–Maxwell and Related Equations

  • Yurii N. Grigoriev
  • Nail H. Ibragimov
  • Vladimir F. Kovalev
  • Sergey V. Meleshko
Part of the Lecture Notes in Physics book series (LNP, volume 806)


This chapter is devoted to a group analysis of the Vlasov–Maxwell and related type equations. The equations form the basis of the collisionless plasma kinetic theory, and are also applied in gravitational astrophysics, in shallow-water theory, etc. Nonlocal operators in these equations appear in the form of the functionals defined by integrals of the distribution functions over momenta of particles.

In the beginning sections the plasma kinetic theory equations are introduced and the way of looking at the symmetries of nonlocal equations is described. Much of the importance of the approach used in this chapter for calculating symmetries stems from the procedure of solving determining equations using variational differentiation. The set of symmetries obtained in the sections that follow comprises symmetries for the Vlasov–Maxwell equations of the non-relativistic and relativistic electron and electron–ion plasmas in both one- and three-dimensional cases, and symmetries for Benney equations. In the concluding sections of this chapter the procedure for symmetry calculation and the renormalization group algorithm go hand in hand to present illustrations from plasma kinetic theory, plasma dynamics, and nonlinear optics, which demonstrate the potentialities of the method in construction of analytic solutions to nonlocal problems of nonlinear physics.


Maxwell Equation Determine Equation Plasma Particle Canonical Operator Lagrangian Velocity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Vlasov, A.A.: The vibrational properties of an electron gas. J. Exp. Theor. Phys. 8(3) (1938) 291–317 (in Russian); see also Sov. Phys. Usp. 10, 721–733 (1968) zbMATHGoogle Scholar
  2. 2.
    Lewak, G.J.: More-uniform perturbation theory of the Vlasov equation. J. Plasma Phys. 3, 243–253 (1969) ADSCrossRefGoogle Scholar
  3. 3.
    Pustovalov, V.V., Chernikov, A.A.: Functional averaging and kinetics of plasma in Lagrangean variables. Preprint No. 171, P.N. Lebedev Physical Institute, AN USSR (1980) (in Russian) Google Scholar
  4. 4.
    Pustovalov, V.V., Romanov, A.B., Savchenko, M.A., Silin, V.P., Chernikov, A.A.: One method for solving the Vlasov kinetic equation. Sov. Phys., Lebedev Inst. Rep. 12, 28–32 (1976) Google Scholar
  5. 5.
    Taranov, V.B.: On the symmetry of one-dimensional high frequency motions of a collisionless plasma. Sov. J. Tech. Phys. 21, 720–726 (1976) Google Scholar
  6. 6.
    Kovalev, V.F., Krivenko, S.V., Pustovalov, V.V.: Group symmetry of the kinetic equations of a collisionless plasma. JETP Lett. 55(4), 256–259 (1992) ADSGoogle Scholar
  7. 7.
    Kovalev, V.F., Krivenko, S.V., Pustovalov, V.V.: Group analysis of the Vlasov kinetic equation, I. Differ. Equ. 29(10), 1568–1578 (1993) MathSciNetGoogle Scholar
  8. 8.
    Kovalev, V.F., Krivenko, S.V., Pustovalov, V.V.: Group analysis of the Vlasov kinetic equation, II. Differ. Equ. 29(11), 1712–1721 (1993) MathSciNetGoogle Scholar
  9. 9.
    Grigor’ev, Yu.N., Meleshko, S.V.: Group analysis of integro-differential Boltzmann equation. Sov. Phys. Dokl. 32, 874–876 (1987) ADSzbMATHGoogle Scholar
  10. 10.
    Volterra, V.: Theory of Functional and of Integral and Integro-Differential Equations. Blackie, London (1929). Edited by Fantappie, L. Translated by Long, M. Also available as: Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations. Dover, New York (1959). Russian translation: Nauka, Moscow (1982) Google Scholar
  11. 11.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.H.: Perturbation methods in group analysis. J. Sov. Math. 55(1), 1450 (1991) CrossRefGoogle Scholar
  12. 12.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.Kh.: Perturbation methods in group analysis. In: Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., vol. 34, pp. 85–147. VINITI, Moscow (1989) (in Russian). J. Sov. Math. 55(1), 1450–1490 (1991) CrossRefGoogle Scholar
  13. 13.
    Dorozhkina, D.S., Semenov, V.E.: Exact solution of Vlasov equations for quasineutral expansion of plasma bunch into vacuum. Phys. Rev. Lett. 81, 2691–2694 (1998) ADSCrossRefGoogle Scholar
  14. 14.
    Kovalev, V.F., Bychenkov, V.Yu., Tikhonchuk, V.T.: Particle dynamics during adiabatic expansion of a plasma bunch. JETP 95(2), 226–241 (2002) ADSCrossRefGoogle Scholar
  15. 15.
    Landau, L.D., Livshitz, E.M.: Course of Theoretical Physics, vol. 2, The Classical Theory of Fields. Nauka, Moscow (1973) Google Scholar
  16. 16.
    Kovalev, V.F., Krivenko, S.V., Pustovalov, V.V.: Symmetry group of Vlasov–Maxwell equations in plasma theory. In: Proceedings of the International Conference “Symmetry in Nonlinear Mathematical Physics”, July 3–8, 1995, Kiev, Ukraina, V. 2. J. Nonlinear Math. Phys. 3(1–2), 175–180 (1996). MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Ibragimov, N.H. (ed.): CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1: Symmetries, Exact Solutions and Conservation Laws (1994); vol. 2: Applications in Engineering and Physical Sciences (1995); vol. 3: New Trends in Theoretical Developments and Computational Methods (1996). CRC Press, Boca Raton zbMATHGoogle Scholar
  18. 18.
    Benney, D.J.: Some properties of long nonlinear waves. Stud. Appl. Math. L11(1), 45–50 (1973) Google Scholar
  19. 19.
    Krasnoslobodtzev, A.V.: Gas dynamic and kinetic analogies in the theory of vertically inhomogeneous shallow water. Trans. Inst. Gen. Phys. USSR Acad. Sci. 18, 33–71 (1989) (in Russian) Google Scholar
  20. 20.
    Kupershmidt, B.A., Manin, Yu.I.: Long-wave equation with free boundary. I. Conservation laws and solutions. Funct. Anal. Appl. 11(3), 188–197 (1977) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kupershmidt, B.A., Manin, Yu.I.: Long-wave equation with free boundary. II. Hamiltonian structure and higher equations. Funct. Anal. Appl. 12(1), 20–29 (1978) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Zakharov, V.E.: Benney equation and quasi-classical approximation in the method of the inverse problem. Funct. Anal. Appl. 14(2), 89–98 (1980) zbMATHCrossRefGoogle Scholar
  23. 23.
    Gibbons, J.: Collisionless Boltzmann equations and integrable moment equations. Physica D3 3(3), 503–511 (1981) MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Ibragimov, N.H., Kovalev, V.F., Pustovalov, V.V.: Symmetries of integro-differential equations: a survey of methods illustrated by the Benney equations. Nonlinear Dyn. 28, 135–153 (2002). Preprint math-ph/0109012 MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Gurevich, A.V., Pitaevski, L.P.: Nonlinear dynamics of a rarefied plasmas and ionospheric aerodynamics. In: Problems of Plasma Theory, vol. 10, pp. 3–87. Nauka, Moscow (1980) (in Russian). Reviews of Plasma Physics, vol. 10. Edited by Acad. Leontovich, M.A. Translated from Russian by Glebov, O. Translation editor: ter Haar, D., Department of Theoretical Physics, University of Oxford, Oxford, England. Published by Consultants Bureau, New York (1986) Google Scholar
  26. 26.
    Kovalev, V.F., Pustovalov, V.V.: Functional self-similarity in a problem of plasma theory with electron nonlinearity. Theor. Math. Phys. 81, 1060–1071 (1990) CrossRefGoogle Scholar
  27. 27.
    Shirkov, D.V.: Several topics on renorm-group theory. In: Shirkov, D.V., Priezzhev, V.B. (eds.) Renormalization Group ‘91, Proc. of Second Intern. Conf., Sept. 1991, Dubna, USSR, pp. 1–10. World Scientific, Singapore (1992) Kovalev, V.F., Krivenko, S.V., Pustovalov, V.V.: The Renormalization group, method based on group analysis. In: Shirkov, D.V., Priezzhev, V.B. (eds.) Renormalization Group ‘91, Proc. of Second Intern. Conf., Sept. 1991, Dubna, USSR, pp. 300–314. World Scientific, Singapore (1992) Google Scholar
  28. 28.
    Kovalev, V.F., Pustovalov, V.V., Shirkov, D.V.: Group analysis and renormgroup symmetries. J. Math. Phys. 39, 1170–1188 (1998). Preprint hep-th/9706056 MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Kovalev, V.F., Shirkov, D.V.: Bogoliubov renormalization group and symmetry of solution in mathematical physics. Phys. Rep. 352(4–6), 219 (2001). hep-th/0001210 MathSciNetADSzbMATHGoogle Scholar
  30. 30.
    Kovalev, V.F., Shirkov, D.V.: The renormalization group symmetry for solution of integral equations. In: Nikitin, A.G. (ed.) Proc. of the 5th Intern. Conf. on Symmetry in Nonlinear Mathematical Physics, Kii’v, Ukraine, June 23–29, 2003. Proc. of the Inst. of Math. of the Natl. Acad. Sci. of Ukraine. Math. and its Appl., vol. 50, Pt. 2, pp. 850–861. Inst. of Math. of NAS Ukraine, Kiïv (2004) Google Scholar
  31. 31.
    Kovalev, V.F., Shirkov, D.V.: Renormgroup symmetry for functionals of boundary value problem solutions. J. Phys. A, Math. Gen. 39, 8061–8073 (2006) MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Kovalev, V.F., Shirkov, D.V.: Renormalization-group symmetries for solutions of nonlinear boundary value problems. Phys.-Usp. 51(8), 815–830 (2008). Preprint arXiv:0812.4821 [math-ph] ADSCrossRefGoogle Scholar
  33. 33.
    Shirkov, D.V.: Renormalization group, invariance principle and functional self-similarity. Sov. Phys. Dokl. 27, 197 (1982) ADSGoogle Scholar
  34. 34.
    Ovsyannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982) zbMATHGoogle Scholar
  35. 35.
    Rudenko, O.V., Soluyan, S.I.: Theoretical Foundations of Nonlinear Acoustics. Consultants Bureau, New York (1977) zbMATHGoogle Scholar
  36. 36.
    Pustovalov, V.V., Silin, V.P.: Nonlinear theory of the interaction of waves in a plasma. In: Proceedings of P.N. Lebedev Physical Institute, AN USSR, vol. 61, pp. 42–283. Nauka, Moscow (1972). English translation in: Skobel’tsyn, D.V. (ed.) Theory of Plasmas. Consultants Bureau, New York (1975) Google Scholar
  37. 37.
    Maksimchuk, A., Flippo, K., Krause, H., et al.: High-energy ion generation by short laser pulses. Plasma Phys. Rep. 30(6), 473–495 (2004) ADSCrossRefGoogle Scholar
  38. 38.
    Kovalev, V.F., Bychenkov, V.Yu: Analytic solutions to the Vlasov equations for expanding plasmas. Phys. Rev. Lett. 90(18), 185004 (2003) (4 pages) ADSCrossRefGoogle Scholar
  39. 39.
    Bychenkov, V.Yu, Kovalev, V.F.: Coulomb explosion in a cluster plasma. Plas. Phys. Rep. 31(2), 178–183 (2005) Google Scholar
  40. 40.
    Bychenkov V.Yu., Kovalev, V.F.: On the maximum energy of ions in a disintegrating ultrathin foil irradiated by a high-power ultrashort laser pulse. Quantum Electron. 35(12), 1143–1145 (2005) ADSCrossRefGoogle Scholar
  41. 41.
    Kovalev, V.F., Popov, K.I., Bychenkov, V.Yu., Rozmus, W.: Laser triggered Coulomb explosion of nanoscale symmetric targets. Phys. Plasmas 14, 053103 (2007) (10 pages) ADSCrossRefGoogle Scholar
  42. 42.
    Kovalev, V.F., Pustovalov, V.V.: Group and renormgroup symmetry of a simple model for nonlinear phenomena in optics, gas dynamics and plasma theory. Math. Comput. Model. 25, 165–179 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Akhmanov, S.A., Sukhorukov, A.P., Khokhlov, R.V.: On the self-focusing and self-chanelling of intense laser beams in nonlinear medium. Sov. Phys. JETP 23(6), 1025–1033 (1966) ADSGoogle Scholar
  44. 44.
    Murakami, M., Kang, Y.-G., Nishihara, K., et al.: Ion energy spectrum of expanding laser-plasma with limited mass. Phys. Plasmas 12, 062706 (2005) (8 pages) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Yurii N. Grigoriev
    • 1
  • Nail H. Ibragimov
    • 2
  • Vladimir F. Kovalev
    • 3
  • Sergey V. Meleshko
    • 4
  1. 1.Inst. Computational TechnologiesRussian Academy of SciencesNovosibirskRussia
  2. 2.Dept. Mathematics & ScienceBlekinge Institute of TechnologyKarlskronaSweden
  3. 3.Inst. Mathematical ModellingRussian Academy of SciencesMoscowRussia
  4. 4.School of Mathematics, Institute of ScienceSuranaree University of Technology (SUT)Nakhon RatchasimaThailand

Personalised recommendations