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Field patterns: a new mathematical object.

Graeme W Milton1, Ornella Mattei1

  • 1Department of Mathematics , University of Utah , Salt Lake City, UT 84112, USA.

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Summary
This summary is machine-generated.

Field patterns in space-time microstructures concentrate along characteristic lines. These patterns evolve independently, akin to separate dimensions, unless nonlinear terms induce interactions.

Keywords:
field patternsspace–time microstructureswave equations

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Area of Science:

  • Physics
  • Applied Mathematics
  • Nonlinear Dynamics

Background:

  • Field patterns are observed in space-time microstructures.
  • Disturbances propagating along characteristic lines typically cascade.
  • Existing models do not fully explain the concentration of disturbances into patterns.

Purpose of the Study:

  • To investigate the conditions under which field patterns form in space-time microstructures.
  • To analyze the behavior and interaction of these field patterns.
  • To explore the potential for interactions via nonlinear terms.

Main Methods:

  • Analysis of characteristic lines in space-time.
  • Mathematical modeling of field pattern formation and evolution.
  • Investigation of transfer matrix eigenvalues and eigenvectors.
  • Consideration of [Formula: see text]-symmetry properties.

Main Results:

  • Field patterns concentrate disturbances along characteristic lines, avoiding cascades.
  • Patterns in one spatial dimension plus time emerge when characteristic slopes are commensurate with the microstructure.
  • Field patterns with spatial shifts evolve independently, resembling separate dimensions.
  • Eigenvalues of the transfer matrix often have unit norm, indicating propagating modes, alongside exponentially growing modes.

Conclusions:

  • Field patterns represent a distinct behavior in space-time microstructures.
  • The independent evolution of patterns suggests a multi-dimensional or multi-component system.
  • Nonlinear terms are hypothesized to introduce interactions between these patterns or components.