随机扰动下周期弹性系统中的安德森局域化
Anderson localization in periodic elastic systems with random perturbations
- School of Mathematics, Jilin University（吉林大学数学学院）
- SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
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中文总结 AI 辅助
本文通过层势与Floquet变换分析随机扰动下亚波长弹性周期系统的安德森局域化，推导特征值渐近公式与共振方程，数值实验验证理论，为弹性超材料局域化提供数学基础。
中文摘要 AI 辅助
本文研究了具有随机扰动的亚波长弹性周期系统中的安德森局域化。对于未扰动系统，我们利用层势技术将特征值问题重新表述为边界积分方程，推导出亚波长特征值的渐近公式，并证明了亚波长带上方存在带隙。对于扰动系统，我们应用Floquet变换获得周期形式，并推导出在一般扰动下确定共振频率的方程。针对扰动的周期单体与二聚体的数值实验与理论预测一致。我们进一步通过增加随机扰动的强度和数量来演示安德森局域化。这些结果为理解弹性超材料中的亚波长局域化提供了数学基础。
英文摘要
This paper investigates Anderson localization in subwavelength elastic periodic systems with random perturbations. For the unperturbed system, we use layer potential techniques to reformulate the eigenvalue problem as boundary integral equations, and derive asymptotic formulas for the subwavelength eigenvalues. For perturbed systems, we apply the Floquet transform to obtain a periodic formulation and derive equations determining the resonant frequencies under general perturbations. Numerical experiments for perturbed periodic monomers and dimers agree with the theoretical predictions. We further demonstrate Anderson localization by increasing the strength and number of random perturbations. These results provide a mathematical foundation for understanding subwavelength localization in elastic metamaterials.
