{"title":"亚波长谐振器系统中的时空波定位","authors":"Habib Ammari, Erik Orvehed Hiltunen, Liora Rueff","doi":"arxiv-2409.10100","DOIUrl":null,"url":null,"abstract":"In this paper we study the dynamics of metamaterials composed of\nhigh-contrast subwavelength resonators and show the existence of localised\nmodes in such a setting. A crucial assumption in this paper is time-modulated\nmaterial parameters. We prove a so-called capacitance matrix approximation of\nthe wave equation in the form of an ordinary differential equation. These\nformulas set the ground for the derivation of a first-principles\ncharacterisation of localised modes in terms of the generalised capacitance\nmatrix. Furthermore, we provide numerical results supporting our analytical\nresults showing for the first time the phenomenon of space-time localised waves\nin a perturbed time-modulated metamaterial. Such spatio-temporal localisation\nis only possible in the presence of subwavelength resonances in the unperturbed\nstructure. We introduce the time-dependent degree of localisation to\nquantitatively determine the localised modes and provide a variety of numerical\nexperiments to illustrate our formulations and results.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Space-Time Wave Localisation in Systems of Subwavelength Resonators\",\"authors\":\"Habib Ammari, Erik Orvehed Hiltunen, Liora Rueff\",\"doi\":\"arxiv-2409.10100\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we study the dynamics of metamaterials composed of\\nhigh-contrast subwavelength resonators and show the existence of localised\\nmodes in such a setting. A crucial assumption in this paper is time-modulated\\nmaterial parameters. We prove a so-called capacitance matrix approximation of\\nthe wave equation in the form of an ordinary differential equation. These\\nformulas set the ground for the derivation of a first-principles\\ncharacterisation of localised modes in terms of the generalised capacitance\\nmatrix. Furthermore, we provide numerical results supporting our analytical\\nresults showing for the first time the phenomenon of space-time localised waves\\nin a perturbed time-modulated metamaterial. Such spatio-temporal localisation\\nis only possible in the presence of subwavelength resonances in the unperturbed\\nstructure. We introduce the time-dependent degree of localisation to\\nquantitatively determine the localised modes and provide a variety of numerical\\nexperiments to illustrate our formulations and results.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.10100\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10100","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Space-Time Wave Localisation in Systems of Subwavelength Resonators
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.