Swift-Hohenberg equation

The Swift-Hohenberg equation (after the two American physicists Jack B. Swift and Pierre C. Hohenberg ) is a mathematical model equation for the investigation of pattern formation processes . A mathematically simplified form of this equation describes the pattern of wrinkling of papillary ridges ( dermatoglyphs ) on fingers, i.e. the pattern of fingerprints , as well as the pattern of the formation of grooves on drying raisins .

the equation

It is a partial differential equation on a real or complex scalar function with two spatial and one temporal arguments: ${\displaystyle \psi }$

${\displaystyle \partial _{t}\psi (x,y,t)=\left[\epsilon -(\Delta +k_{\text{crit}}^{2})^{2}\right]\cdot \psi -R(\psi )}$.

Are there

• ${\displaystyle \partial _{t}}$the partial derivative with respect to time
• the parameter is the analogue of the temperature in the Bénard experiment ${\displaystyle \epsilon }$
• ${\displaystyle \Delta }$the Laplace operator
• ${\displaystyle k_{\text{crit}}}$a critical circular wavenumber
• ${\displaystyle R(\psi )}$a nonlinear function with . ${\displaystyle R(0)=0}$

Of particular interest is the appearance of after a sufficiently long time , i.e. H. the stable solutions of the equation, if any are ever reached. ${\displaystyle \psi (x,y)}$ ${\displaystyle t}$

Homogeneous solution

For is the stable solution of the equation. ${\displaystyle \epsilon <0}$ ${\displaystyle \psi \equiv 0}$

Critical point

The behavior around the critical point becomes evident after a Fourier transformation of the linear component of the equation: ${\displaystyle \epsilon =0}$

${\displaystyle \partial _{t}{\tilde {\psi }}(k,t)=[\epsilon -(k_{\text{crit}}^{2}-k^{2})^{2}]\cdot {\tilde {\psi }}}$

• In this case , the amplitudes converge towards zero for all wave numbers , so no pattern is formed. ${\displaystyle \epsilon <0}$ ${\displaystyle {\tilde {\psi }}}$
• If , the amplitudes of some supercritical wave numbers increase. The supercritical wave numbers form a circle with the radius . A pattern is formed with the wavelength . ${\displaystyle \epsilon >0}$ ${\displaystyle k_{\text{crit}}}$ ${\displaystyle 2\pi /k_{\text{crit}}}$

Overly critical behavior

The supercritical behavior for is determined by the shape of . Similar to the Bénard experiment, the solutions are typically rolls or hexagonal patterns. ${\displaystyle \epsilon >0}$ ${\displaystyle R(\psi )}$

literature

• MC Cross and PC Hohenberg, Rev. Mod. Phys. 65: 851 (1993).
• J. Swift (Department of Physics, University of Texas, Austin), PC Hohenberg (Bell Laboratories, Murray Hill; Physics Department, Technical University of Munich): Hydrodynamic fluctuations at the convective instability . Phys. Rev. A 15, 319-328 (1977)

Individual evidence

1. ↑ J. Swift, P. Hohenberg: Hydrodynamic fluctuations at the convective instability. In: Physical Review A. 15, 1977, p. 319, doi : 10.1103 / PhysRevA.15.319 .
2. ↑ Holger Dambeck: Mathematicians explain patterns of fingerprints. Spiegel Online, February 4, 2015, accessed February 5, 2015 . 
3. ↑ Norbert Stoop, Romain Lagrange, Denis Terwagne, Pedro M. Reis, Jörn Dunkel: Curvature-induced symmetry breaking determines elastic surface patterns. In: Nature Materials. 14, 2015, p. 337, doi : 10.1038 / NMAT4202 .