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Fractional dynamics in nonlinear magnetic metamaterials
Journal of Magnetism and Magnetic Materials ( IF 2.5 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.jmmm.2020.167573
Mario I. Molina

We examine the existence of nonlinear modes and their temporal dynamics, in arrays of split-ring resonators, using a fractional extension of the Laplacian in the evolution equation. We find a closed-form expression for the dispersion relation as a function of the fractional exponent as well as an exact expression for the critical coupling between rings, beyond which no fractional magnetoinductive wave can exist. We also find the low-lying families of bulk and surface nonlinear modes and their bifurcation diagrams. Here the phenomenology is similar for all exponents and resembles what has been observed in other discrete evolution equations, such as the DNLS. The propagation of an initially localized magnetic excitation is always ballistic, with a `speed' that is computed in exact form as a function of the fractional exponent. For a given exponent, it increases with an increase in coupling up to a critical coupling value, beyond which the ballistic speed could diverge inside the fractional interval $[0,1]$. Examination of the modulational instability shows that it tends to increase with an increase in the fractional exponent, where the decay proceeds via the formation of filamentary structures that merge eventually and form pure radiation. The dynamical selftrapping around an initially localized excitation increases with the fractional exponent, but it also shows a degree of trapping in the linear limit. This trapping increases with a decrease in the exponent and can be explained by near-degeneracy considerations.

中文翻译:

非线性磁性超材料的分数动力学

我们在演化方程中使用拉普拉斯算子的分数扩展来检查裂环谐振器阵列中非线性模式的存在及其时间动态。我们找到了作为分数指数函数的色散关系的封闭形式表达式,以及环之间临界耦合的精确表达式,超出该值将不存在分数磁感应波。我们还发现了体和表面非线性模式的低层族及其分叉图。这里的现象学对于所有指数都是相似的,并且类似于在其他离散演化方程(例如 DNLS)中观察到的现象。初始局部磁激励的传播始终是弹道的,其“速度”以分数指数的函数形式以精确形式计算。对于给定的指数,它随着耦合的增加而增加,直至达到临界耦合值,超过该值,弹道速度可能会在分数区间 $[0,1]$ 内发散。对调制不稳定性的检查表明,它倾向于随着分数指数的增加而增加,其中衰减通过形成最终合并并形成纯辐射的丝状结构进行。初始局部激发周围的动态自陷随分数指数的增加而增加,但它也显示出线性限制中的一定程度的自陷。这种俘获随着指数的减小而增加,并且可以通过接近退化的考虑来解释。超过该值,弹道速度可能会在分数区间 $[0,1]$ 内发散。对调制不稳定性的检查表明,它倾向于随着分数指数的增加而增加,其中衰减通过形成最终合并并形成纯辐射的丝状结构进行。初始局部激发周围的动态自陷随分数指数的增加而增加,但它也显示出线性限制中的一定程度的自陷。这种俘获随着指数的减小而增加,并且可以通过接近退化的考虑来解释。超过该值,弹道速度可能会在分数区间 $[0,1]$ 内发散。对调制不稳定性的检查表明,它倾向于随着分数指数的增加而增加,其中衰减通过形成最终合并并形成纯辐射的丝状结构进行。初始局部激发周围的动态自陷随分数指数的增加而增加,但它也显示出线性限制中的一定程度的自陷。这种俘获随着指数的减小而增加,并且可以通过接近退化的考虑来解释。初始局部激发周围的动态自陷随分数指数的增加而增加,但它也显示出线性限制中的一定程度的自陷。这种俘获随着指数的减小而增加,并且可以通过接近退化的考虑来解释。初始局部激发周围的动态自陷随分数指数的增加而增加,但它也显示出线性限制中的一定程度的自陷。这种俘获随着指数的减小而增加,并且可以通过接近退化的考虑来解释。
更新日期:2021-04-01
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