I have a lot of catching up to do. I want to share a bunch of work by Steve Huntsman. I’ll start with some older material. A bit of this may be ‘outdated’ by his later work, but I figure it’s all worth recording.
One goal here is to define ‘stripes’ for the Kuramoto–Sivashinky equation in a way that lets us count them, their births, and their mergers, and so on. We need a good definition to test the conjectures I made in Part 1.
While I originally formulated my conjectures for the ‘integral form’ of the
Steve has mostly been working with the derivative form:
so you can assume that unless I say otherwise. He’s using periodic boundary conditions such that
for some length The length depends on the particular experiment he’s doing.
First, a plot of stripes. It looks like here:
Births and deaths are shown as green and red dots, respectively. But to see them, you may need to click on the picture to enlarge it!
According to my conjecture there should be no red dots. The red dots at the top and the bottom of the image don’t count: they mostly arise because this program doesn’t take the periodic boundary conditions into account. There are two other red dots, which are worth thinking about.
Nice! But how are stripes being defined here? He describes how:
The stripe definition is mostly pretty simple and not image processy at all, and the trick to improve it is limited to removing little blobs and is easily explained.
Let be the solution to the KSE. Then let
where is the average integer offset (maybe I’m missing a minus sign a la ) that maximizes the cross-correlation between and . Now anywhere exceeds its median is part of a stripe.
The image processing trick is that I delete little stripes (and I use what image processors would call 4-connectivity to define simply connected regions—this is the conservative idea that a pixel should have a neighbor to the north, south, east, or west to be connected to that neighbor, instead of the aggressive 8-connectivity that allows NE, NW, SE, SW too) whose area is less than 1000 grid points. So it uses lots of image processing machinery to actually do its job, but the definition is simple and easily explained mathematically.
An obvious fix that removes the two nontrivial deaths in the picture I sent is to require a death to be sufficiently far away from another stripe: here I am guessing that the characteristic radius of a stripe will work just fine.