The Kuramoto–Sivashinsky Equation (Part 4)

Here is some more work by Cheyne Weis. Last time I explained that Cheyne and Steve Huntsman were solving the ‘derivative form’ of the Kuramoto–Sivashinsky equation, namely this:

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

Above is one of Steve’s pictures of a typical solution with its characteristic ‘stripes’. Cheyne started trying to identify these stripes as locations where du/dx \le c for some cutoff c, for example c = -0.7. He made some nice 3d views of du/dx illustrating the problems with this. As he explains:

So there is the tradeoff that if I make the cutoff too high, the sections that look greenish are getting identified as stripes. If I make it too low, the points near where two stripes merge disappear. I attached some 3D plots showing the landscape of du/dx reiterating this point.

The disappearance of the stripes as they merge is unavoidable to a certain extent. The plots in the PowerPoint show how even when there is no cutoff, there is some gap between the merging stripes. There may be some characteristic length scale needed to qualify them as merging.

You can click on these pictures to make them bigger.

2 Responses to The Kuramoto–Sivashinsky Equation (Part 4)

  1. dwinsemius says:

    Is there any way to see (as in define energy and momentum akin to our versions of physical reality) this as a 2D (one x and one t dimension) quantized system where the “continuous segments” are real particles and the shorter segments are virtual particles?

    • John Baez says:

      I believe the shorter segments are really stripes that get absorbed by other stripes shortly after they’re born, before they’re able to grow to their full size.

      But to your main question: there could be a stochastic process where particles are born and merge, which closely mimics the qualitative and even the quantitative behavior of the Kuramoto–Sivashinsky equation. Theodore Kolokolnikov has made progress in this direction here:

      The Kuramoto–Sivashinsky equation (part 6)

      His equation is actually deterministic! It’s possible a bit of stochasticity would help, because the Kuramoto–Sivashinsky equation is chaotic (for large lengths L). But he’s already able to get approximately the right look:

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