Structure of shock waves in planar motion of plasma

Abstract

The mathematical question of the existence of structure for shock waves in planar motion of plasma is reduced to finding heteroclinic orbits of a system of five ordinary differential equations, which depend on five viscosity parameters. It is shown that the system is gradient-like and has four rest points. An explicit bound is obtained for the set of the bounded complete solutions, and an isolating neighbourhood for this system is constructed. Then using Conley theory we prove that the fast and slow shock waves, of arbitrary strength, always possess structure. Using a limiting technique, the switch-on and switch-off shock structures are obtained as a topological limit of the fast and slow shock structures, respectively. Furthermore, we show that most of the intermediate shocks, in the absence of the electric field, admit structure. The stability of all of these structures is also shown.