Communications in Mathematical Sciences

Volume 9 (2011)

Number 1

Kerr–Debye relaxation shock profiles for Kerr equations

Pages: 1 – 31



Denise Aregba-Driollet (Institut de Mathématiques de Bordeaux,Université de Bordeaux, Talence, France)

Bernard Hanouzet (Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, France)


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.


nonlinear hyperbolic problems, relaxation, shock profiles, Kerr-Debye model

2010 Mathematics Subject Classification

35L65, 35L67, 35Q60

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