Long Time Asymptotics of Solutions to the Anharmonic Oscillator Model From Nonlinear Optics

: The anharmonic oscillator model desribing the propagation of electromagnetic waves in an exterior domain containing a nonlinear dielectric medium is investigated. Local decay of the electromagnetic field for t !1 in the charge free case is shown. If the potential of the oscillating particles is in addition convex it is shown that for arbitrary initial states the electromagnetic field converges for t ! 1 to an electrostatic field determined by the initial data and the prescribed external current. Key words: nonlinear optics, Maxwell's equations, exterior boundary-value-problem, asymptotic behaviour, AMS subject classification: 35Q60, 35L40, 78A35. 1 Introduction The subject of this paper is the anharmonic oscillator model from nonliear optics consisting of Maxwell's equations @ t E = curl H \Gamma @ t ~ P \Gamma j; @ t H = \Gamma curl E; (1.1) on IR + \Theta\Omega coupled with the equation ff@ 2 t P+ @ t P+r P V (x; P) = flE (1.2) on IR + \Theta G. The initial boundary..