arXiv 2005.000862020-05nlin.PS

非线性磁性超材料中的分数阶动力学

Fractional dynamics in nonlinear magnetic metamaterials

Mario I. Molina

备注：: 7 pages, 8 figures
DOI：: 10.1016/j.jmmm.2020.167573

摘要：

我们研究了非线性模式的存在及其时间动态，在阵列的开口环谐振器，使用分数扩展的拉普拉斯算子的演化方程。我们找到了一个封闭形式的表达式的色散关系作为分数指数的函数，以及环之间的临界耦合的精确表达式，超过它没有分数磁感应波可以存在。我们还发现了体和表面非线性模态的低域族及其分岔图。这里的现象是类似的所有指数，并类似于已观察到的其他离散演化方程，如DNLS。初始局部磁激励的传播总是弹道式的，其“速度”以精确形式计算为分数指数的函数。对于一个给定的指数，它随着耦合的增加而增加，直到临界耦合值，超过该临界耦合值，弹道速度可能在分数间隔$[0,1]$内发散。调制不稳定性的检查表明，它往往会随着分数指数的增加而增加，其中衰变通过最终合并并形成纯辐射的非线性结构的形成而进行。初始局域激发周围的动态自陷随分数指数增加，但它也显示出一定程度的线性极限的陷阱。这种捕获随着指数的减小而增加，并且可以通过近简并考虑来解释。

英文摘要：

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.

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