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Invariant solutions of the equations of a multicomponent charged-particle beam

  • V. A. Syrovoi
Article
  • 29 Downloads

Abstract

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.

Keywords

Periodic Solution Arbitrary Function Vector Versus Spherical Geometry Group Property 
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.

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References

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    L. V. Ovsyannikov, “Groups and invariant-group solutions of differential equations,” DAN SSSR, vol. 118, no. 3, 1958.Google Scholar
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    L. V. Ovsyannikov, “The group properties of an equation of nonlinear thermal conductivity,” DAN SSSR, vol. 125, no. 3, 1959.Google Scholar
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    L. V. Ovsyannikov, The Group Properties of Differential Equations [in Russian], Izd. SO AN SSSR, Novosibirsk, 1962.Google Scholar
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    V. A. Syrovoi, “Invariant-group solutions of the equations of a one-dimensional stationary chargedparticle beam,” PMTF, no. 4, 1962.Google Scholar
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    V. A. Syrovoi, “Invariant-group solutions of the equations of a three-dimensional stationary charged-particle beam,” PMTF, no. 3, 1963.Google Scholar
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    V. A. Syrovoi, “Invariant-group solutions of the equations of a nonstationary charged-particle beam,” PMTF, no. 1, 1964.Google Scholar
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    V. A. Syrovoi, “On some new solutions that can be obtained by means of invariant transformations,” PMTF, no. 3, 1965.Google Scholar

Copyright information

© The Faraday Press, Inc. 1967

Authors and Affiliations

  • V. A. Syrovoi
    • 1
  1. 1.Moscow

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