In this paper we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localised modes in such a setting. A crucial assumption in this paper 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-principles characterisation of localised modes in terms of the generalised capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space-time localised waves in a perturbed time-modulated metamaterial. Such spatio-temporal localisation is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localisation to quantitatively determine the localised modes and provide a variety of numerical experiments to illustrate our formulations and results.

翻译：本文研究了由高对比度亚波长谐振子构成的超材料动力学，并证明了在此类系统中存在局域化模式。本文的一个关键假设是材料参数的时间调制。我们证明了波动方程的一种所谓电容矩阵近似，其形式为常微分方程。这些公式为基于广义电容矩阵的局域化模式第一性原理表征奠定了基础。此外，我们提供了支持分析结果的数值计算，首次展示了在扰动时间调制超材料中时空局域波的现象。这种时空局域化仅在未扰动结构中存在亚波长共振时才可能实现。我们引入了时间依赖的局域化度来定量确定局域模式，并通过多种数值实验来阐释我们的公式与结果。