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

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 have a transverse magnetic polarization. These are often referred to as breathers. They are given as a Fourier series in $y$ and $t$ with the leading frequency $ω$ 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.