1D Scattering through time dependent media with memory

TL;DR

This paper develops a mathematical framework for scattering in 1+1 wave equations with media that have memory, extending classical scattering theory to time-dependent, non-local permittivities, and supports numerical solution schemes.

Contribution

It introduces an operator-valued scattering matrix for media with memory, generalizing traditional scattering matrices to time-dependent, non-local permittivities in wave equations.

Findings

Constructed a scattering matrix with operator-valued entries.

Provided a mathematical basis for recent numerical methods.

Presented a numerical scheme for solving the wave equation with memory.

Abstract

We construct a scattering matrix with operator valued entries describing solutions to the 1+1 wave equation where permittivities has memory and depends on time and space. It is the analogue of the scattering matrix for spatially localised perturbations where the entries are functions of frequency and appear as Fourier multipliers in solutions of the wave equation. This provides a mathematical explanation of the numerical construction in the recent paper by Horsley et al. The appendix by Zhen Huang and Maciej Zworski presents a numerical scheme for solving the wave equation considered in this article.