Polychromatic Localized Waves with Complex Frequencies in Nonlinear Maxwell Equations with Material Dispersion

TL;DR

This paper proves the existence of localized, polychromatic wave solutions in nonlinear Maxwell equations with dispersive media, revealing complex frequency decay and constructing examples related to surface plasmons in waveguides.

Contribution

It introduces a novel method to establish existence of polychromatic localized waves in nonlinear dispersive media using spectral analysis and operator pencils.

Findings

Existence of polychromatic solutions with complex frequencies in nonlinear Maxwell equations.

Construction of solutions as Fourier series with decay rates and frequencies.

Application to waveguides with interfaces supporting nonlinear surface plasmons.

Abstract

We study the existence of polychromatic solutions of cubically nonlinear Maxwell equations in the whole space and with dispersive media, i.e., with a time delayed polarization. Due to the complex nature of the dielectric function, the frequencies are complex, resulting in a decay in time. The geometry is that of a waveguide in $x$ with the propagation direction being $y$ and the solutions are localized in $x$ and TM-polarized. These are often referred to as breathers. They are given as a Fourier series in $y$ and $t$ with the leading frequency $\omega$ being an eigenvalue of a corresponding operator pencil on $\mathbb{R}$ (in the $x$ variable). Each term in the series corresponds to a different temporal decay rate or a different frequency. The series is constructed iteratively via a sequence of linear ordinary differential equations. Our general result provides the existence under some assumptions on the spectrum and on estimates of the resolvent of the corresponding linear operator. We also produce an example of a waveguide given by the interface of two spatially homogeneous physically relevant media for which these assumptions are satisfied. For such an interface setting the constructed solutions correspond to nonlinear polychromatic surface plasmons.