Field Patterns: A New Mathematical Object

TL;DR

This paper introduces the concept of field patterns in space-time microstructures, describing their formation, properties, and potential interactions, with implications for wave dynamics and PT-symmetry.

Contribution

It defines and analyzes field patterns as a new mathematical object in space-time microstructures, exploring their behavior, independence, and potential interactions under nonlinear extensions.

Findings

Field patterns occur when characteristic slopes are commensurate with microstructure.

Different field patterns evolve independently, akin to separate dimensions.

PT-symmetry leads to propagating and exponentially growing modes.

Abstract

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 multicomponent 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 multidimensional space that one can consider them to live, or between the different field components of the multicomponent potential if one views them that way. As a result of PT-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.