Abstract
The solution of the exact, nonlinear one-dimensional Vlasov equation with a space- and time-dependent electric field is reduced to the solution of a nonlinear, first-order ordinary differential equation with two subsidiary equations. The reduction holds for any electric field nonlinear in the spatial coordinate or for a subclass of electric fields linear in the spatial coordinate and is equivalent to the solution of a generalized Bernstein–Greene–Kruskal (BGK) Vlasov equation with a velocity-dependent Lorentz force. The Lie method for the solution of differential equations invariant under a transformation group has been used to calculate the group generator, the canonical variables, and the generalized BGK Vlasov equation. Analytical forms for the functional dependence of the Vlasov equation one-particle distribution function are given.
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