Multiscale Scattering in Nonlinear Kerr-Type Media

TL;DR

This paper introduces a multiscale method for solving nonlinear Helmholtz problems with oscillating coefficients, combining localized orthogonal decomposition and adaptive nonlinearity approximation, supported by rigorous analysis and numerical validation.

Contribution

It develops a novel multiscale approach that handles nonlinear Kerr-type media without structural assumptions on coefficients, with proven convergence and practical numerical results.

Findings

Method achieves accurate error estimates

Approach is applicable to complex oscillatory media

Numerical examples confirm theoretical predictions

Abstract

We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously analyze the method in terms of well-posedness and convergence properties based on suitable assumptions on the initial data and the discretization parameters. Numerical examples illustrate the theoretical error estimates and underline the practicability of the approach.