I coded up a simple dynamical system with the following rules (loosely motivated by theory of motion of spikes in reaction-diffusion systems, see e.g. appendix of this paper, as well as this paper):

• insert a particle if inter-particle distance is more than some maxdist
• merge any two particles that collide
• otherwise evolve particles according to the ODE

Here, is a Green’s function that satisfies

inside the interval with Neumann boundary conditions . Explicitly,

and

where sign(0) is taken to be zero so that

In particular, for large , one has

and

Here are some of the resulting simulations, with different (including complex ). This is mainly just for fun but there is a wide range of behaviours. In particular, I think the large-lambda limit should be able to capture analogous dynamics in the Keller-Siegel model with logistic growth (Hillen et. al.), see e.g. figures in this paper.

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2 Responses to The Kuramoto–Sivashinsky Equation (Part 6)

There might be some system like this that closely emulates the Kuramoto–Sivashinsky equation. I guess that’s part of your point. In the Kuramoto–Sivashinsky equation a new stripe seems to appear whenever there’s enough room. I want to figure out much room that is, and how it’s related to the wavelength of the most unstable mode of the linearized equation, namely

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You need the word 'latex' right after the first dollar sign, and it needs a space after it. Double dollar signs don't work, and other limitations apply, some described here. You can't preview comments here, but I'm happy to fix errors.

I like the look of these pictures a lot!

There might be some system like this that closely emulates the Kuramoto–Sivashinsky equation. I guess that’s part of your point. In the Kuramoto–Sivashinsky equation a new stripe seems to appear whenever there’s enough room. I want to figure out much room that is, and how it’s related to the wavelength of the most unstable mode of the linearized equation, namely

Perhaps there is a renormalization group argument/universality class lurking around the statistics of stripes.