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

TL;DR

The article proves the existence of polychromatic, exponentially damped breather-like solutions to cubically nonlinear Maxwell equations with dispersive, time-delayed polarization in a waveguide. It introduces a TM reduction, a hierarchical linear-ODE reformulation, and an operator-pencil spectral framework, establishing existence under resolvent-estimate assumptions. A concrete Lorentz-interface model with finite memory is analyzed to verify the assumptions, and numerical experiments illustrate the polychromatic solution and its convergence. The results illuminate how non-self-adjointness and finite memory enable polychromatic localization and nonlinear surface-plasmon phenomena at material interfaces, with potential extensions to multiple interfaces and higher dimensions.

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 $ω$ 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.

Figures (6)

• Figure 1: Schematic of the set $\mathbf{S}$ of mixed integer multiples $\omega_0^{(n,\nu)}$ for $\nu\leq 5$.
• Figure 2: Numerical computations for the spectrum of the truncated Lorentz model.
• Figure 3: Convergence of the numerically computed eigenvalues for the truncated Lorentz model.
• Figure 4: Numerical approximation of the first and second component of $\varepsilon \varphi$ and the partial sum $\psi^{(M)}$ with $M=15$, see \ref{['E:part-sum']}, and with $\varepsilon=20$. Here $\varphi$ is the eigenfunction \ref{['E:efn']} of the truncated operator ${\mathcal{L}}_{3}^T$ with parameters given by \ref{['eqn:parameters']}. Insets show regions, in which the polychromatic solution deviates the most from the linear eigenfunction.
• Figure 5: The $L^2$-norms over $\mathbb R\times (0,2\pi)$ of the numerically computed approximations $u^\nu(x,y):=\sum_{|n|\leq\nu} u^{n,\nu}(x)e^{{\rm i} nky}$, i.e., all terms that decay at rate $\nu\omega_I$ evaluated at time $t=0$, again for the set of parameters \ref{['eqn:parameters']} and $\varepsilon=20$ (left) or $\varepsilon=0.5$ (right). Note that the plot is semilogarithmic, such that the dependence on $\nu$ is exponential.

Theorems & Definitions (51)

• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Theorem 2
• Lemma 3.1
• proof