On the properties of affine solutions of cold plasma equations

We study the affine solutions of the equations of plane oscillations of cold plasma, which, under the assumption of electrostaticity, correspond to the Euler–Poisson equations in the repulsive case. It is proved that the zero equilibrium state of the cold plasma equations, both with and without the assumption of electrostaticity, is unstable in the class of all affine solutions. It is also shown that an arbitrary perturbation of an axially symmetric electrostatic solution leads to a finite time blow-up.

Keywords

cold plasma, Euler–Poisson equations, quasilinear system, affine solutions, blow up

2010 Mathematics Subject Classification

Primary 35Q60. Secondary 34Mxx, 35L60.

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The authors’ work was supported by the Moscow Center for Fundamental and Applied Mathematics.

Received 30 November 2022

Received revised 31 May 2023

Accepted 31 May 2023

Published 7 December 2023