Swift–Hohenberg equation

The Swift–Hohenberg equation (named after Jack B. Swift and Pierre Hohenberg) is a partial differential equation noted for its pattern-forming behaviour. It takes the form

where u = u(x, t) or u = u(x, y, t) is a scalar function defined on the line or the plane, r is a real bifurcation parameter, and N(u) is some smooth nonlinearity.

The equation is named after the authors of the paper,[1] where it was derived from the equations for thermal convection.

The webpage of Michael Cross[2] contains some numerical integrators which demonstrate the behaviour of several Swift–Hohenberg-like systems.

Applications

Geometric Measure Theory

In 2009 Ruggero Gabbrielli[3] published a way to use the Swift-Hohenberg equation to find candidate solutions to the Kelvin Problem on minimal surfaces.[4][5]

References
  1. J. Swift,P.C. Hohenberg (1977). "Hydrodynamic fluctuations at the convective instability". Phys. Rev. A. 15: 319–328. doi:10.1103/PhysRevA.15.319.CS1 maint: uses authors parameter (link)
  2. Java applet demonstrations
  3. Gabbrielli, Ruggero. "Ruggero Gabbrielli - Google Scholar Citations". scholar.google.com.
  4. Gabbrielli, Ruggero (1 August 2009). "A new counter-example to Kelvin's conjecture on minimal surfaces". Philosophical Magazine Letters. 89 (8): 483–491. doi:10.1080/09500830903022651. ISSN 0950-0839.
  5. Freiberger, Marianne (24 September 2009). "Kelvin's bubble burst again | plus.maths.org". Plus Magazine. University of Cambridge. Retrieved 4 July 2017.


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