2016/11/18 by Graeme W. Milton, Ornella Mattei · 1 voice · 2 citations
Computer Science · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Nonlinear Dynamics and Pattern Formation #Theoretical and Computational Physics #math-ph #physics.optics
paper · pdf · doi:10.1098/rspa.2016.0819
arxiv published 2016/11/18 · openalex publication_date 2017/02/01 · arxiv updated 2017/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.