Internal free boundary problem for cold plasma equations

For the system of cold plasma equations describing the motion of electrons in the field of stationary ions, we consider the Riemann problem posed at an impenetrable interface between two media. These media differ in the magnitude of the constant ion field. The interface between the media is assumed to be free. Its position is determined from the generalized Rankine-Hugoniot conditions and the stability condition, that is, the intersection of Lagrangian particle trajectories at the interface.

💡 Research Summary

The paper investigates an internal free‑boundary problem for the one‑dimensional cold‑plasma equations, which model the motion of electrons under the electrostatic field of stationary ions. The authors consider two half‑spaces separated by an impenetrable interface at (x=\Phi(t)). Each half‑space has a constant background ion density, denoted (n_{-}>0) for (x<\Phi(t)) and (n_{+}>0) for (x>\Phi(t)). The initial data are of Riemann type: the electron velocity (V) and electric field (E) have a jump at the origin, while the density follows from the relation (\rho=n_{\pm}-E_{x}). The position of the interface (\Phi(t)) is not prescribed; it must be determined as part of the solution.

The governing system in dimensionless form is
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