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Plasma Kinetic Theory: Vlasov–Maxwell and Related Equations

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Symmetries of Integro-Differential Equations

Part of the book series: Lecture Notes in Physics ((LNP,volume 806))

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Abstract

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.

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Notes

  1. 1.

    Usually it is assumed for gas particles that the energy of their interaction is small compared to their kinetic energy. Up to the order of magnitude the latter can be estimated as κ T, where T is the temperature and κ is the Boltzmann constant. For charged plasma particles the energy of interaction is of the order of e 2 N 1/3, where N −1/3 is the mean distance between particles, e is a charge and N is the number of particles in a unit volume. Hence the plasma demonstrates the gas property provided that

    $$e^2N^{1/3}\ll \kappa T.$$

    This inequality holds for all real plasmas.

  2. 2.

    Equation (4.1.1) is approximate, as it neglects collisions of plasma particles. In view of particle collisions their motion becomes correlated. This effect leads to appearance of non-zero term in the right-hand side of (4.1.1), the so-called collision integral. However, the explicit form of the collision integral depends on particular conditions defined by the plasma properties in every concrete situation, and we will not discuss them here. In many particular problems collision effects can be neglected.

  3. 3.

    Frequently the six operators specifying hyperbolic and circular rotations in (c 2 t,x k) and (x j,x k) planes, respectively (j,k=1,2,3;r=(x 1,x 2,x 3)), are written in a universal form using the operators M μ ν , where M 0k =iB0k and M jk =iR jk . The three operators (M 23,M 31,M 12) are components of the vector-operator M=[r×P].

  4. 4.

    For more details we refer the reader to [24].

  5. 5.

    For simplicity these equations are omitted here.

  6. 6.

    This is representable as the integral \(\int_{0}^{1}n(x,\vartheta)\,\mathrm{d}\cos\vartheta\) of the kinetic equation solution n(x,ϑ).

  7. 7.

    Here the bottom index specifies on a corresponding tensor component, instead of designating a derivative.

  8. 8.

    The term ‘quasi-Chaplygin media’ is used in the discussion of nonlinear phenomena developing in accordance with the mathematical scenario for the Chaplygin gas, i.e., the gas with a negative adiabatic exponent. At first glance, such a model looks like the standard model of gas dynamics, but it corresponds to the negative first derivative of the ‘pressure’ with respect to the ‘density.’ A characteristic feature of quasi-Chaplygin media is a universal mathematical form of various nonlinear effects accompanying the development of an instability.

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Authors and Affiliations

  1. Inst. Computational Technologies, Russian Academy of Sciences, Pr. Lavrentjeva 6, 630090, Novosibirsk, Russia

    Prof. Yurii N. Grigoriev

  2. Dept. Mathematics & Science, Blekinge Institute of Technology, 371 79, Karlskrona, Sweden

    Prof. Nail H. Ibragimov

  3. Inst. Mathematical Modelling, Russian Academy of Sciences, Miusskaya Square 4A, 125047, Moscow, Russia

    Dr. Vladimir F. Kovalev

  4. School of Mathematics, Institute of Science, Suranaree University of Technology (SUT), 30000, Nakhon Ratchasima, Thailand

    Sergey V. Meleshko

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  1. Prof. Yurii N. Grigoriev
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  2. Prof. Nail H. Ibragimov
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  3. Dr. Vladimir F. Kovalev
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  4. Sergey V. Meleshko
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Correspondence to Yurii N. Grigoriev .

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Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V. (2010). Plasma Kinetic Theory: Vlasov–Maxwell and Related Equations. In: Symmetries of Integro-Differential Equations. Lecture Notes in Physics, vol 806. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3797-8_4

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