The Kuramoto–Sivashinsky Equation (Part 3)

I’ve been getting a lot of help from Steve Huntsman and also Cheyne Weis, who is a physics grad student at the University of Chicago. You can see a lot, but far from all, of Steve’s work as comments on part 1. Here are some things Cheyne has been doing.

Cheyne started out working with the ‘derivative form’ of the Kuramoto–Sivashinsky equation, meaning this:

u_t + u_{xx} + u_{xxxx} + u u_x = 0

and he soon noticed what Steve made clear in the image above: the ‘stripes’ in solutions of this equation aren’t ‘bumps’ (regions where u is large) but regions where the solution is rapidly changing from positive to negative. This suggests a way to define stripes: look for where du/dx < c for some negative c. It seems c = -0.7 is a pretty good choice.

I thought maybe it would be better to use the derivative of the PDE’s solution (du/dx) to define the stripes. You can find an image of this in the attached PowerPoint.

The second slide has another image where the lines represent the minima of du/dx (as a function of x) that are below a certain threshold c. You can see these lines appearing and combining as apparent in Thien An’s animation. Hopefully this is some progress on the definition of a “bump”. If you agree, I could use this to test some of your other conjectures.

Here are the result for a range of alternative choices of c. The problem, if we’re seeking a definition of ‘stripe’ where stripes never die as time passes, is the presence of short ‘ministripes’ that die shortly after they appear. What’s really going on, I believe, is that when small stripes merge with larger ones, the derivative du/dx becomes smaller in absolutely value, thus going above the cutoff c. In short, merging is being misinterpreted as death.

2 Responses to The Kuramoto–Sivashinsky Equation (Part 3)

  1. dwinsemius says:

    Of those 4 lowest images of minima, the “backbone” of the plots are basically the same, but the shorter terminating arcs are placed similarly but are progrssively shorter as c increases.

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