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 ) }  }


\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} }


\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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