0 1 2 3 4 5 4 3 2 1 0

and a small bump like 0 1 1+d 0, where d is small. Combining them, we might get

0 2 3+d 3 4 5 4 3 2 1 0

If d is just above zero, the 3+d is a maximum. If it is just below zero, it disappears. This is not a numerical problem: refining the x and t resolution doesn’t help. On the other hand, looking the gradient near the point of disappearance does tell you where to find the big bump. So I think you can join things up that way, with lines of constant t.

I haven’t implented this, except by printing out images and joining some gaps by hand. It is clear that the branching pattern is very unbalanced. When two subtrees join, they each have a number of tips, say i and n-i. In simple models of branching processes, the probabilities of 1:(n-1), 2:(n-2), … (n-1):1 are all equal to 1/(n-1). The Kuramotoâ€“Sivashinksy branching pattern favours the extremes. In particular, there are lots of 1:(n-1) and (n-1):1. I would like someone to prove a theorem about this!

Something else I noticed when trying the integral version with large L. If you look at for fixed , and at low resolution so that the bumps and dips are lost, the result looks (to me) like a Brownian bridge (https://en.wikipedia.org/wiki/Brownian_bridge). I have a glimmer of an idea about why that happens. If is a solution, then so is for any constant . So in a range which is big compared the bumps, but small compared to L, there is nothing to stop drifting away, except the gentle long range tug that comes from having to be periodic.

]]>In this picture, the “particles” look more like “particle-antiparticle” bound states. The blue stripes are the zeros in u where the sign of u is changing from positive to negative. If you zoom in on a bright merger event, it looks like the innermost particle and antiparticle annihilate, and the outermost particle and antiparticle recombine into a new bound state. The whole picture looks like a vacuum breaking down and forming a condensate.

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