Milton Graeme W, Mattei Ornella
Department of Mathematics , University of Utah , Salt Lake City, UT 84112, USA.
Proc Math Phys Eng Sci. 2017 Feb;473(2198):20160819. doi: 10.1098/rspa.2016.0819.
Field patterns occur in space-time microstructures such that a disturbance propagating along a characteristic line does not evolve into a cascade of disturbances, but rather concentrates on a pattern of characteristic lines. This pattern is the field pattern. In one spatial direction plus time, the field patterns occur when the slope of the characteristics is, in a sense, commensurate with the space-time microstructure. Field patterns with different spatial shifts do not generally interact, but rather evolve as if they live in separate dimensions, as many dimensions as the number of field patterns. Alternatively one can view a collection as a multi-component potential, with as many components as the number of field patterns. Presumably, if one added a tiny nonlinear term to the wave equation one would then see interactions between these field patterns in the multi-dimensional space that one can consider them to live, or between the different field components of the multi-component potential if one views them that way. As a result of [Formula: see text]-symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. There are also modes that blow up exponentially with time.
场模式出现在时空微观结构中，使得沿特征线传播的扰动不会演变成一系列扰动，而是集中在特征线的一种模式上。这种模式就是场模式。在一个空间方向加时间的情况下，当场模式出现时，在某种意义上，特征线的斜率与时空微观结构是相称的。具有不同空间偏移的场模式通常不会相互作用，而是各自独立演化，就好像它们存在于不同的维度中，维度数量与场模式的数量相同。或者，人们可以将一组场模式视为一个多分量势，其分量数量与场模式的数量相同。据推测，如果在波动方程中添加一个微小的非线性项，那么在人们可以认为它们所处的多维空间中，就会看到这些场模式之间的相互作用，或者如果以这种方式看待它们，就会看到多分量势的不同场分量之间的相互作用。由于[公式：见原文]对称性，适当定义的转移矩阵的许多复特征值具有单位范数，因此相应的特征向量对应于传播模式。也存在随时间指数增长的模式。
