Plasma kinetic theory: Vlasov-maxwell and related equations

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. © 2010 Springer Science+Business Media B.V.