A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity

Abstract

The present paper is devoted to the study of a class of nonlinear Helmholtz equations that is essentially used in mathematical models for the theoretical and numerical analysis of scattering and radiation effects. While well-known works relate to geometrically simple domains (supporting the nonlinearity) and selected nonlinearities, the new aspects lie in the transition to more generally shaped, two- or three-dimensional objects, to more general nonlinearities (including saturation), and in the possibility of an efficient numerical approximation of the electromagnetic fields and derived quantities (such as energy, transmission coefficient, etc.).The paper describes and investigates an approach that consists in transforming the original full-space transmission problem for a nonlinear Helmholtz equation 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 nonlinear problem is equivalent to the original one and is uniquely solvable under appropriate conditions. In addition, the effect of truncation of the DtN operator on the resulting solution is investigated. It is shown that the corresponding sesquilinear form satisfies a parameter-uniform inf–sup condition, that the modified nonlinear problem has a unique solution, and that the solution error caused by the truncated DtN operator can be estimated in dependence on the truncation parameter.