Kerr–Debye relaxation shock profiles for Kerr equations

Abstract

Abstract. The electromagnetic wave propagation in a nonlinear medium can be described by a Kerr model in the case of an instantaneous response of the material, or by a Kerr-Debye model if the material exhibits a finite response time. Both models are quasilinear hyperbolic, and the Kerr-Debye model is a physical relaxation approximation of the Kerr model. In this paper we characterize the shocks in the Kerr model for which there exists a Kerr-Debye profile. First we consider 1D models for which explicit calculations are performed. Then we determine the plane discontinuities of the full vector 3D Kerr system and their admissibility in the sense of Liu and in the sense of Lax. Finally we characterize the large amplitude Kerr shocks giving rise to the existence of Kerr-Debye relaxation profiles.

Godunov scheme for Maxwell’s equations with Kerr nonlinearity

“…Those conditions are very different from the 6 × 6 case, where for a 2-shock the requirement λ 1 (u − ) ≤ σ imposes a sign condition on d + , see [1]. Here, this sign condition no longer exists.…”

“…This result is proved in [1] for the particular case of system (1.8). The proof of proposition 3.4 follows the same lines so we omit it.…”

“…The admissible shocks of the Kerr system have already been studied in [1]. For the sake of completenes those results are briefly recalled here.…”

“…In [1], we have proved that f is a twice differentiable concave increasing function with respect to d 2 ≥ 0 with…”

“…For a fixed left state u − = (D − , H 3,− ) we can proceed as in [1] to determine the related Hugoniot set, that is the set of all right states u + satisfying the RankineHugoniot relations, which read here:…”