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Wavenumber selection via spatial parameter jump.

Arnd Scheel1, Jasper Weinburd2

  • 1School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN 55455, USA scheel@umn.edu.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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Summary
This summary is machine-generated.

This study investigates the Swift-Hohenberg equation with spatial inhomogeneity, revealing a new transition from homogeneous states to striped patterns. The resulting stripe wavenumbers are restricted to a narrow band, unlike in homogeneous cases.

Keywords:
inhomogeneous medianormal formsspatial dynamicsstrain–displacementturing instability

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Area of Science:

  • Nonlinear dynamics
  • Pattern formation
  • Mathematical physics

Background:

  • The Swift-Hohenberg equation models pattern formation near instability onset.
  • Spatial inhomogeneity can significantly alter pattern formation dynamics.
  • Understanding transitions between homogeneous and patterned states is crucial in nonlinear systems.

Purpose of the Study:

  • To analyze the Swift-Hohenberg equation with a spatial jump-type inhomogeneity.
  • To identify and characterize steady-state solutions representing spatial transitions.
  • To investigate the impact of inhomogeneity on the selection of pattern wavenumbers.

Main Methods:

  • Application of normal form theory.
  • Analysis using spatial dynamics.
  • Numerical continuation methods.
  • Direct simulations for stability analysis.

Main Results:

  • Existence of steady-state solutions exhibiting a spatial transition from homogeneous to striped patterns.
  • The wavenumbers of the emerging stripes are confined to a narrow band.
  • The width of this wavenumber band scales linearly with the magnitude of the spatial jump.

Conclusions:

  • Spatial inhomogeneity severely restricts pattern selection in the Swift-Hohenberg equation.
  • The linear scaling of the wavenumber band width contrasts with the square-root scaling in homogeneous systems.
  • The findings have implications for understanding pattern stability and formation in spatially varying environments.