The Kuramoto–Sivashinsky Equation (Part 6)

guest post by Theodore Kolokolnikov

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

$\displaystyle{ x'_k(t) = \sum_{j=1}^N G_x(x_k, x_j) }$

Here, $G$ is a Green’s function that satisfies

$G_{xx}-\lambda^2 G = -\delta(x,y)$

inside the interval $[-L,L]$ with Neumann boundary conditions $G_x(\pm L, y)=0$. Explicitly,

$\displaystyle{ G(x,y) = \frac{\cosh((x+y)\lambda)+\cosh((2L-|x-y|) \lambda )}{2 \lambda \sinh(2L \lambda ) } }$

and

$\displaystyle{ G_x(x,y)= \frac{\sinh((x+y) \lambda )+ \sinh((|x-y|-2L) \lambda ) \mbox{sign}(x-y) } {2 \sinh(2L\lambda)} }$

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

$\displaystyle{ G_x(y,y) := \frac{G_x(y^+,y)+ G_x(y^-,y)}{2} }$

In particular, for large $\lambda$, one has

$\displaystyle{ G(x,y)\sim\frac{e^{-\lambda | x-y|}}{2\lambda} }$

and

$\displaystyle{ G_x(x,y)\sim-\frac{e^{-\lambda | x-y|}} {2} \mbox{sign}(x-y), ~~\lambda \gg 1 }$

Here are some of the resulting simulations, with different $\lambda$ (including complex $\lambda$). 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.

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

1. John Baez says:

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

$2 \sqrt{2} \, \pi$

2. Steve Huntsman says:

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

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