# Breather solutions to nonlinear Maxwell equations with retarded material laws

## TL;DR

This paper proves the existence of localized, time-periodic solutions called breathers in nonlinear Maxwell equations with retarded material responses, using variational methods and spectral gap assumptions.

## Contribution

It introduces a variational approach to find breather solutions in Maxwell's equations with retarded nonlinearities, under spectral gap conditions.

## Key findings

- Existence of breathers in Kerr-type optical materials with retarded nonlinear response.
- Identification of spectral gap conditions for the linear operator.
- Examples of material coefficients with spectral gaps, including nonperiodic materials.

## Abstract

We consider Maxwell's equations for Kerr-type optical materials, which are magnetically inactive and have a nonlinear response to electric fields. This response consists of a linear plus a cubic term, which are both inhomogeneous with bounded coefficients. The cubic term is temporally retarded while the linear term has instantaneous and retarded contributions. For slab waveguides we show existence of breathers, which are time-periodic, real-valued solutions that are localized in the direction perpendicular to the waveguide, and moreover they are traveling along one direction of the waveguide. We find these breathers using a variational method which relies on the assumption that an effective operator related to the linear part of Maxwell's equations has a spectral gap about $0$. We also give examples of material coefficients, including nonperiodic materials, where such a spectral gap is present.