Field Patterns: A New Mathematical Object

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.

31 pages, 19 figures

References in corpus (10)

• Optical Hyperlens: Far-field imaging beyond the diffraction limit
• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Synthetic gauge fields in synthetic dimensions
• Synthetic dimensions in integrated photonics: From optical isolation to 4D quantum Hall physics
• Nonlinear switching and solitons in PT-symmetric photonic systems
• Quantum simulation of an extra dimension
• Photonic gauge potential in a system with a synthetic frequency dimension
• Coherent Cavitation in the Liquid of Light
• Defect Modes and Homogenization of Periodic Schrödinger Operators
• High frequency homogenization for travelling waves in periodic media

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• Soft Temporal Switching of TL Parameters: Wave-field, Energy Balance, Applications
• Temporal boundaries in electromagnetic materials
• Generalized FDTD Scheme for the Simulation of Electromagnetic Scattering in Moving Structures
• On the band gap universality multiphase laminates and its applications
• Electromagnetic fields in a time-varying medium: Exceptional points and operator symmetries
• Field patterns without blow up
• Elastic wavepackets crossing a space-time interface
• Field patterns: a new type of wave with infinitely degenerate band structure
• Edge Waves and Transmissions for Temporal Laminates and Imperfect Chiral Interfaces
• A unifying perspective on linear continuum equations prevalent in science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time
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