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.

Shock structure for electromagnetic waves in bianisotropic, nonlinear materials

“…This classification depends on how the shock speed relates to the characteristic speeds in the material, which in turn depend on the field strengths on both sides of the shock. These shock conditions can probably be improved with the help of Conley's index theory, as in the works of Smoller [1994] and Farjami and Hesaaraki [1998].…”

“…The conditions on λ 1,2 above can be written in terms of the shock speed v and the characteristic wave speeds c 1,2 as These expressions constitute the entropy conditions for electromagnetic, plane shock waves. The nomenclature “fast shock” and “slow shock” is in accordance with the works of Gvozdovskaya and Kulikovskii [1999] and Farjami and Hesaaraki [1998] and “intermediate shock” is from the work of Farjami and Hesaaraki [1998]. Note that the fast and the slow shock are closely connected to the ordinary and extraordinary rays for anisotropic materials [see Kong , 1986, pp.…”

“…Farjami and Hesaaraki [1998] and ''intermediate shock'' is from the work of Farjami and Hesaaraki [1998]. Note that the fast and the slow shock are closely connected to the ordinary and extraordinary rays for anisotropic materials [see Kong, 1986, pp.…”

“…Perhaps the most familiar kind of “electromagnetic” shock wave is in the field of magnetohydrodynamics, from which we give only a few references [ Bekefi and Brown , 1966; Conley and Smoller , 1974; Farjami and Hesaaraki , 1998; Germain , 1960; Grabbe , 1984; Landau et al , 1984, pp. 245–253].…”

“…This is the shock structure problem, which was introduced by Gel'fand [1963], and is given an extensive treatment by Smoller [1994]. A thorough treatment of this problem in magnetohydrodynamics is given by Farjami and Hesaaraki [1998], and a recent paper deals with the structure of electromagnetic shock waves in anisotropic ferromagnetic media [ Gvozdovskaya and Kulikovskii , 1999].…”