Field patterns without blow up
Field patterns without blow up
复制标题
没有爆炸的场模式
DOI：
10.1088/1367-2630/aa847d
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发表时间：
2017
影响因子：
3.3
通讯作者：
G. Milton
中科院分区：
文献类型：
--
作者：
O. Mattei;G. Milton
场模式，首先由米尔顿和Mattei（2017 Proc. R. Soc. A 473 20160819）是一种新型波，沿着特征线的有序图案沿着传播，这些特征线出现在特定的时空微结构中，其在一个空间维度和时间上的几何形状在某种程度上与特征线的斜率相称。特别是，在米尔顿和Mattei（2017 Proc. R. Soc. A 473 20160819），作者提出了两个时空几何形状的例子，其中出现场图案：它们是两相微结构，其中一种材料的矩形时空夹杂物嵌入另一种材料中。在足够长的时间间隔之后，场模式在时间和空间上都具有局部周期性。这使得人们可以只专注于解决场模式存在的离散网络上的问题，并定义一个合适的传递矩阵，该矩阵在给定某个时间的解的情况下，在一个时间段后提供解。对于上述微结构，该非对称传递矩阵的许多特征值具有单位范数，因此对应的特征向量对应于传播模式。然而，也有随时间呈指数增长的模式，以及随时间呈指数下降的模式。问题是，是否存在这样的时空微观结构，使得传递矩阵只在单位圆上具有本征值，从而不存在增长模式（爆破模式）？答案就在这里，在这里我们看到，某些时空棋盘具有这样的性质，即在一定的材料参数范围内，所有的模式都是传播模式。有趣的是，当不存在爆破时，由瞬时扰动在某一点产生的波看起来像是带有振荡波尾流的激波，非常明显的是，其振幅在远离波前的地方不趋于零。
Field patterns, first proposed by the authors in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819), are a new type of wave propagating along orderly patterns of characteristic lines which arise in specific space-time microstructures whose geometry in one spatial dimension plus time is somehow commensurate with the slope of the characteristic lines. In particular, in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819) the authors propose two examples of space-time geometries in which field patterns occur: they are two-phase microstructures in which rectangular space-time inclusions of one material are embedded in another material. After a sufficiently long interval of time, field patterns have local periodicity both in time and space. This allows one to focus only on solving the problem on the discrete network on which a field pattern lives and to define a suitable transfer matrix that, given the solution at a certain time, provides the solution after one time period. For the aforementioned microstructures, many of the eigenvalues of this -symmetric transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. However, there are also modes that blow up exponentially with time coupled with modes that decrease exponentially with time. The question arises as to whether there are space-time microstructures such that the transfer matrix only has eigenvalues on the unit circle, so that there are no growing modes (modes that blow-up)? The answer is found here, where we see that certain space-time checkerboards have the property that all the modes are propagating modes, within a certain range of the material parameters. Interestingly, when there is no blow-up, the waves generated by an instantaneous disturbance at a point look like shocks with a wake of oscillatory waves, whose amplitude, very remarkably, does not tend to zero away from the wave front.