Abstract
We study the influence of the factor of electron-ion collisions on the solution of the Cauchy problem in the one-dimensional relativistic model of cold plasma and show that, depending on their intensity and initial data, two scenarios are possible: either the solution remains smooth and stabilizes to a stationary state, or during a finite time the oscillations blowup. In contrast to the nonrelativistic model, when exact conditions can be obtained separating the two behaviors, in a much more complicated relativistic situation, it turns out to be possible to analytically estimate from below the time during which the existence of a smooth solution and the guaranteed number of oscillations during this time. In addition, we show that in contrast to the relativistic case without taking into account collisions, when oscillations corresponding to arbitrarily small deviations from the zero equilibrium position blow up, the presence of electron collisions can suppress the blowup of sufficiently small oscillations. Further, based on the analysis of characteristics, a numerical algorithm is constructed, the order of accuracy of which is determined only by the smoothness of the initial data. Numerical experiments are presented to illustrate the theoretical results. The initial conditions are chosen as reasonably as possible from the point of view of full-scale physical experiments.
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References
Alexandrov, A.F., Bogdankevich, L.S., Rukhadze, A.A.: Principles of Plasma Electrodynamics. Springer series in electronics and photonics, Springer, Berlin Heidelberg (1984)
Bellan, P.M.: Fundamentals of Plasma Physics. Cambridge University Press, Cambridge (2006)
Booker, H.G.: Cold Plasma Waves. Springer, Berlin (2004)
Chizhonkov, E.V.: Mathematical Aspects of Modelling Oscillations and Wake Waves in Plasma. CRC Press, Cambridge (2019)
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. The 4th Edition, (Berlin: Springer, 2016)
Davidson, R.C.: Methods in nonlinear plasma theory. Academic Press, New York (1972)
Dawson, J.M.: Nonlinear electron oscillations in a cold plasma. Phys. Rev. 113(2), 383–387 (1959)
Engelberg, S., Liu, H., Tadmor, E.: Critical thresholds in Euler–Poisson equations. Ind. Univ. Math. J. 50, 109–157 (2001)
Frolov, A.A., Chizhonkov, E.V.: Influence of electron collisions on the breaking of plasma oscillations. Plasma Phys. Rep. 44, 398–404 (2018)
Ginzburg, V. L.: Propagation of Electromagnetic Waves in Plasma ( Pergamon: New York, 1970)
Infeld, E., Rowlands, G., Skorupski, A.A.: Analytically solvable model of nonlinear oscillations in a cold but viscous and resistive plasma. Phys. Rev. Lett. 102(1–4), 145005 (2009)
Kahaner, D., Moler, C., Nash, S.: Numerical Methods and Software. Prentice-Hall International Inc, New York (1989)
Rozanova, O.S., Chizhonkov, E.V.: On the conditions for the breaking of oscillations in a cold plasma. Z. Angew. Math. Phys. 72, 13 (2021). https://doi.org/10.1007/s00033-020-01440-3
Rozanova, O.S., Chizhonkov, E.V.: On the existence of a global solution of a hyperbolic problem. Doklady Math. 101, 254–256 (2020)
Rozanova, O.S., Chizhonkov, E.V., Delova, M.I.: Exact thresholds in the dynamics of cold plasma with electron-ion collisions. AIP Conf. Proc. (2020). https://doi.org/10.1063/5.0033619
Verma, P.S., Soni, J.K., Segupta, S., Kaw, P.K.: Nonlinear oscillations in a cold dissipative plasma. Phys. Plasmas 17(1-4), 044503 (2010)
Sheppard, C.J.R.: Cylindrical lenses—focusing and imaging: a review [Invited]. Appl. Opt. 52, 538–545 (2013)
Schultz, M.H.: Spline Analysis. Prentice-Hall International Inc, New York (1973)
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Partially supported by the Moscow Center for Fundamental and Applied Mathematics.
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Rozanova, O.S., Chizhonkov, E.V. Stabilization and blowup in the relativistic model of cold collisional plasma. Z. Angew. Math. Phys. 72, 184 (2021). https://doi.org/10.1007/s00033-021-01615-6
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DOI: https://doi.org/10.1007/s00033-021-01615-6