## 1. Introduction

[2] Electromagnetic waves propagating in an instantaneously reacting material can be modeled with a system of quasi-linear partial differential equations. It is well known that such a model can exhibit shock solutions, i.e., solutions that become discontinuous in finite time even if the initial/boundary data are smooth. This poses severe problems for numerical methods, such as finite difference schemes, which are often based on the assumption of continuous and differentiable solutions.

[3] In order to overcome this problem, we can model the material on a finer scale, which requires a denser grid and thus increases the memory demands and the computation time. Another approach is to develop more powerful numerical methods, which can handle discontinuous solutions. The development of these numerical schemes benefits from an understanding of the propagation of shock waves; for instance, Godunov's scheme is based on the solution of Riemann's problem [*Godlewski and Raviart*, 1996; *Godunov*, 1959], where the shock wave is generated by discontinuous initial data. A variation of Godunov's scheme is Glimm's scheme, which is used to show global existence of solutions to certain systems of equations [*Glimm*, 1965; *Hörmander*, 1997].

[4] The aim of this paper is to increase the understanding of electromagnetic shock waves, modeled with the Maxwell equations. Mainly using techniques from the work of *Sjöberg* [2000], we analyze the wave propagation in bianisotropic materials, i.e., materials with different properties for different polarizations of the waves, and a possible coupling between the electric and the magnetic field [*Kong*, 1986, p. 7]. This adds insight not only to the numerical treatment of electromagnetic waves in complicated materials, but also provides some physical intuition.

[5] The Maxwell equations can be considered as a hyperbolic system of conservation laws. A good introduction to the numerical approximation of such systems is given by *Godlewski and Raviart* [1996], which introduces the analytical theory as well as some common schemes in one and two spatial dimensions. There is presently not a good mathematical understanding of systems of conservation laws in several dimensions, but some general references are the works of *Godlewski and Raviart* [1996], *Hörmander* [1997], *Dafermos* [2000], *Serre* [2001], and *Taylor* [1996].

[6] 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]. Electromagnetic shock waves in isotropic media have previously been treated theoretically [*Bloom*, 1993, and references therein; see *Landau et al.*, 1984, pp. 388–391]. Recently, a few papers on experiments concerning electromagnetic shock waves have been published [*Dolan*, 1999; *Brooker et al.*, 1999; *Branch and Smith*, 1996]. In continuum mechanics, Maugin has recognized the similarity between shock waves and phase transition fronts as singular sets in irreversible motion, with a dissipation related to the power expanded by a driving force on the singularity set [see *Maugin*, 1998, 2000].

[7] In this paper, we study when the shock waves can be defined as the limit of continuous traveling wave solutions to an approximate problem, where the discontinuity is smoothed over a small region. 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].

[8] This paper is organized as follows. In section 2 we introduce the Maxwell equations and the constitutive relations used to model the electromagnetic waves, as well as the general form of the entropy condition. In section 3 we present the vanishing viscosity method of smoothing the solutions of quasi-linear hyperbolic equations. The consequences of the vanishing viscosity method for traveling waves are studied in sections 4 and 5, where we show that there exists three kinds of electromagnetic shock waves: the fast, the slow, and the intermediate shock wave. In section 6 we also mention another form of discontinuous solutions, contact discontinuities, which cannot be analyzed with the vanishing viscosity method for traveling waves. However, they exist only under the condition of linear degeneracy, and we present this condition and its opposite, genuine nonlinearity, in section 6. The different kinds of shock waves are illustrated with phase portraits of a certain system of ordinary differential equations in section 7, and some concluding remarks are made in section 8.