A non‐isentropic Euler–Poisson model for a collisionless plasma

We analyse transonic solutions of the one‐dimensional Euler–Poisson model for a collisionless gas of charged particles in the non‐isentropic steady‐state case. The model consists of the conservation of mass, momentum and energy equations. The electric field is modelled self‐consistently (Coulomb field). Boundary conditions on the particle density and particle temperature are imposed. The analysis is based on representing solutions piecewise as orbits in the particle‐density‐electric‐field phase plane and connecting the orbit segments by the jump and entropy conditions. We characterize the set of all solutions of the Euler–Poisson problem. In particular, we show that, depending upon the length of the interval on which the boundary value problem is posed, fully subsonic, one‐shock and (in certain cases) two‐shock transonic and smooth transonic solutions exist. Also, numerical computations illustrating the structure of the solutions are reported.