Stationary and oscillating solutions of the ionization equations

In this work, a number of mathematical problems of the theory of ionization as applied to processes in stationary plasma thrusters are solved. Two main mathematical models of ionization, hydrodynamic and kinetic, are considered. The focus is on the existence of ionization oscillations (breathing modes). Based on a one-dimensional hydrodynamic model, a boundary value problem for stationary ionization equations is solved. Its unique solvability and the absence of breathing modes are proved in the case of sign-definite velocities of atoms and ions. In a practically important case when the ion velocity in the flow region has a single zero with a positive derivative, it is proved that the stationary boundary value problem has a countable number of solutions and a necessary and sufficient condition for the existence of breathing modes is formulated. A numerical algorithm for analysis of breathing modes is proposed. An analytical solution of the ionization equations is given in the case of constant atom and ion velocities, and the resulting formulas are applied to the analytical solution of the Cauchy problem and to boundary value and mixed problems in simplest domains. In the case of a one-dimensional kinetic model of ionization, the existence of breathing modes is shown numerically and a brief analysis of the results obtained is conducted.