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

TL;DR

This paper extends a method for analyzing scattering and radiation effects in nonlinear Helmholtz equations, focusing on more complex geometries, nonlinearities, and efficient numerical solutions using boundary-value problem transformations.

Contribution

It introduces a generalized approach transforming nonlinear Helmholtz problems into boundary-value problems with nonlocal operators, enabling efficient numerical analysis for complex geometries and nonlinearities.

Findings

The transformed boundary-value problem is equivalent to the original nonlinear Helmholtz problem.

The approach ensures unique solutions under certain conditions.

Investigation of DtN operator truncation impacts on numerical solutions.

Abstract

The present work describes some extensions of an approach, originally 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 new aspects lie on the transition to more generally shaped, two- or three-dimensional objects, which no longer necessarily have to be represented in terms 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 paper advocates an approach that consists in transforming the original full-space problem for a nonlinear Helmholtz equation (as the simplest model) 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 is equivalent to the original one and can be solved uniquely under suitable conditions. Morever, the impact of the truncation of the DtN operator on the resulting solution is investigated, so that the way to the numerical solution by appropriate finite element methods is available.