Invariant solutions of the equations of a multicomponent charged-particle beam
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The concept of the invariant-group solution (H-solution) was introduced and a general method for obtaining it was developed in [1–3]. The group properties of the equations of a monoenergetic charged-particle beam with the same value and sign of the specific chargeη, assuming univalency of the velocity vector V, were studied in [4–6], where all essentially different H-solutions were also constructed. Below, the results of [4–6] are extended to the case of a beam in the presence of a fixed background of density ρ0 (§1), and also to the case of multivelocity (V is an s-valued function) and multicomponent beams (i.e., beams formed by particles of several kinds) (§2). A number of analytic solutions that describe some nonstationary processes in devices with plane, cylindrical, and spherical geometry —among them a continuous periodic solution for a plane diode with a period determined by the background density -are obtained in §1. A transformation that contains arbitrary functions of time and preserves Vlasov's equations is given (§2). The equations studied can be treated as the equations of a rarefied plasma in the magnetohydrodynamic approximation, when the pressure gradients are negligible as compared with forces of electromagnetic origin.
KeywordsPeriodic Solution Arbitrary Function Vector Versus Spherical Geometry Group Property
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