Swift-Hohenberg Equation
This applet uses a generalized Swift-Hohenberg model to
illustrate some of the general principles of pattern formation. The
equation used is
and is discussed in Section 5.6.1. In the above
equation, ψ(x,y,t) is the pattern forming field, a function
of two extended space variables x,y and time
and ε is the control parameter. For g1 =
0 the equation reduces to the original Swift-Hohenberg equation,
see Section 5.1.
The plot shows the field ψ on a rainbow color scale with
blue the minimum value and red the maximum value. These maximum and
minimum values are displayed at the top of the plot. Alternatively,
if Plot FFT is set to Yes the magnitude of the Fourier
transform of ψ is plotted, with the origin of the wave
vector at the center of the plot. Random initial conditions are used,
and the evolution is reinitialized whenever Reset is hit.
Here are some suggestions for investigations you might do:
- Swift-Hohenberg equation (set g1 = 0)
- What happens for ε < 0 ?
- Carefully study what happens for a small positive value
of ε such as ε = 0.3. View both the
field ψ and its Fourier transform.
- Generalized Swift-Hohenberg equation (set g1 = 1, for example)
- Investigate what happens for both negative and positive values
of ε. Note that the system shows hysteresis, and you
should study the behavior on increasing and decreasing &epsilon
as the pattern evolves.
Last modified July 5, 2009