Space–time wave localization in systems of subwavelength resonators

In this article, we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localized modes in such a setting. A crucial assumption in this article is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principle characterization of localized modes in terms of the generalized capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space–time localized waves in a perturbed time-modulated metamaterial. Such spatio-temporal localization is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localization to quantitatively determine the localized modes and provide a variety of numerical experiments to illustrate our formulations and results.