Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-06T09:18:35.901Z Has data issue: false hasContentIssue false
  • English
  • Français

The general Lie group and similarity solutions for the one-dimensional Vlasov–Maxwell equations

Published online by Cambridge University Press:  13 March 2009

Dana Roberts
Dana Roberts
Affiliation:
Department of Electrical Engineering, Washington University, St Louis, Missouri 63130
Get access Rights & Permissions [Opens in a new window]

Abstract

The general Lie point transformation group and the associated reduced differential equations and similarity forms for the solutions are derived here for the coupled (nonlinear) Vlasov–Maxwell equations in one spatial dimension. The case of one species in a background is shown to admit a larger group than the multi-species case. Previous exact solutions are shown to be special cases of the above solutions, and many of the new solutions are found to constrain the form of the distribution function much more than, for example, the BGK solutions do. The individual generators of the Lie group are used to find the possible subgroups. Finally, a simple physical argument is given to show that the asymptotic solution (t→∞) for a one-species, one-dimensional plasma is one of the general similarity solutions.

Type
Research Article
Information
Journal of Plasma Physics , Volume 33 , Issue 2 , April 1985 , pp. 219 - 236
Copyright
Copyright © Cambridge University Press 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abraham-Shrauner, B. 1984 a Phys. Fluids, 27, 197.CrossRefGoogle Scholar
Abraham-Shrauner, B. 1984 b J. Plasma Phys. 32, 197.Google Scholar
Abraham-Shrauner, B. 1985 J. Math. Phys. (In press.)Google Scholar
Anderson, R. L. & Ibragimov, N. H. 1979 Lie-Backlund Transformations in Applications. SIAM.CrossRefGoogle Scholar
Baranov, V. B. 1976 Soviet Phys. Tech. Phys. 21, 724.Google Scholar
Bernstein, I. B., Green, J. M. & Kruskal, M. D. 1957 Phys. Rev. 108, 546.CrossRefGoogle Scholar
Bluman, G. W. & Cole, J. D. 1974 Similarity Methods for Differential Equations. Springer.Google Scholar
Burgan, J. R., Feix, M. R., Fijalkow, E. & Munier, A. 1983 J. Plasma Phys. 29, 139.CrossRefGoogle Scholar
Cohen, A. 1911 An Introduction to the Lie Group Theory of One-Parameter Groups. Heath.Google Scholar
Lewis, H. R. & Leach, P. G. L. 1982 J. Math. Phys. 23, 2371.CrossRefGoogle Scholar
Lewis, H. R. & Symon, K. R. 1984 Phys. Fluids, 27, 192.CrossRefGoogle Scholar
Zel'Dovich, Ya. B. & Raizer, Yu. P. 1967 Physics of Shock Waves and High Temperature Hydrodynamic Phenomena. Academic.Google Scholar