Formulation and investigation of a mathematical model for resonance radiation on media with compactly supported nonlinearities

Abstract

The present work outlines some extensions of an approach, developed by V.V. Yatsyk and the author, for the theoretical and numerical analysis of scattering and radiation effects on infinite plates with cubically polarized layers. The focus of these modifications lies on the transition to more generally shaped, two- or three-dimensional objects, which no longer necessarily have to be represented as a Cartesian product of real intervals, to more general nonlinearities (including saturation) and the possibility of an efficient numerical approximation of the electromagnetic fields and derived quantities (such as energy, transmission coefficient, etc.). The present work advocates an approach that consists in transforming the original full-space problem for a system of nonlinear partial differential equations into an equivalent boundary value problem on a bounded domain by means of a nonlocal Dirichlet-to-Neumann (DtN) operator. It is shown that the transformed problem can be solved uniquely under suitable conditions, so that the way to the numerical solution by appropriate finite element methods in conjunction with localization techniques of the DtN operator is available.