A differential equation with behavior resembling the "random" look of CA rules 30, 45, or 73.

Pictured below is the partial differential equation that Wolfram considered in NKS to best resemble the behavior of the most interesting CA rules; below that are a few other pde's that Wolfram said somewhat resemble the most interesting CA behavior. Beside those pde's is the basic output of Rule 30. (All pics taken from the NKS website.) As others have commented, these pde's really don't resemble the output of CA's like Rule 30, and seem more to be just a combination of overlapping waves in various patterns. The distinctive random appearance of Rule 30 seems not just quantitatively different, but qualitatively distinct.

I was looking for a pde that more clearly captured that distinctive "random" look of rules 30, 45, 73, or 110...

Rule 30

Here is the first result I got that seemed qualitatively much more like CA Rule 30 than the other pde's.

[Graphics:HTMLFiles/index_7.gif]

The above is the plot of:

$NDSolve[{∂_ (t, t) y[x, t] ∂_xy[x, t] * ∂_ (x, x) y[x, t] - y[x, t]^ ... ), y[5, t] y[-5, t], Derivative[0, 1][y][x, 0] 0}, y[x, t], {x, -5, 5}, {t, 0, 2}]$

That is, for the function y(x,t), the second derivative wrt t equals the first derivative wrt x times the second derivative wrt x, minus y^3.

The initial and boundary conditions are basically the same as in NKS: a gaussian bump at t=0, initially flat in the y-t plane, wrapped at x=5=-5.

The basic CA rules can be seen as descriptions of curves in the x-y plane along with rules for whether the function at that point should increase or decrease (eg, black-black-white means the right shoulder of a hump). The most interesting CA rules seem to be the most asymmetrical, so the equations here were chosen to asymmetically affect curves. So they tend to pull one side of a hump down and the other side up but also shear the middle of an s-shaped curve. Additionally, the two factors, the first and second derivatives (or, below, the first dervative and y[x,t]) tend to affect different regions of a curve differently.

As can be seen below, the amplitude oscillates strongly.

[Graphics:HTMLFiles/index_4.gif]

The -y^3 component was intended to damp oscillations and constrain the range (though one may see at the bottom of this page that it may do more than that). In an attempt to verify that this graph doesn't owe its interesting qualities to singularities, discontinuities, or Mathematica calculation errors, the damping factor was increased to -y^15, which allows for a bit more nuance to appear.

$NDSolve[{∂_ (t, t) y[x, t] ∂_xy[x, t] * ∂_ (x, x) y[x, t] - y[x, t]^ ... , t], Derivative[0, 1][y][x, 0] 0}, y[x, t], {x, -5, 5}, {t, 0, 5}, MaxSteps30000]$

[Graphics:HTMLFiles/index_17.gif]

[Graphics:HTMLFiles/index_20.gif]

[Graphics:HTMLFiles/index_26.gif]

Here, RHS = first derivative wrt x * y - y^7

$NDSolve[{∂_ {t, 2} y[x, t] ∂_ {x, 1} y[x, t] * y[x, t] - y[x, t]^7 ... 1;Derivative[0, 1][y][x, 0] 0},  y[x, t], {x, -5, 5}, {t, 0, 20}]$

[Graphics:HTMLFiles/index_a9.gif]

The following are based on related equations, targeting specific regions of a curve :

[Graphics:HTMLFiles/index_a11.gif]

[Graphics:HTMLFiles/index_a12.gif]

[Graphics:HTMLFiles/index_a13.gif]

[Graphics:HTMLFiles/index_a14.gif]

[Graphics:HTMLFiles/index_a15.gif]

[Graphics:HTMLFiles/index_a16.gif]

$NDSolve[{∂_ {t, 2} y[x, t] ∂_ {x, 1} y[x, t] * y[x, t], y[ ... ivative[0, 1][y][x, 0] 0}, y[x, t], {x, -3, 3}, {t, 0, 10}, MaxSteps30000]$

[Graphics:HTMLFiles/index_a19.gif]

$DSolve[∂_ {t, 2} y[x, t]  -(y[x, t]), y[x, t], {x, t}]$

$DensityPlot[Cos[t] Exp[-x^2] + Sin[t] * x,  {x, -10, 10}, {t, 0, 10}, PlotPoints200, MeshFalse]$

[Graphics:HTMLFiles/index_a24.gif]

As can be seen above, the -y^3 factor may have an important role in generating some of the complexity of the equation, much as Wolfram's forced sin(t) boundary condition was intended to.