Chaotic Fronts: The Kuramoto-Sivashisky equation

Instructions

This Java applet (If you don't see anything above, please install
Java 1.4) describes the spatio-temporal evolution of the Kuramoto-Sivashinsky equation and higher order variants. It is regulated by a number of control buttons, of which the main button on top regulate the type of equation.

Standard case is the Kuramoto-Sivashinsky equation. This can be directly exchanged ion the Kuramoto-S button by a version where the instability term is changed from a -Laplacian to a Laplacian(Laplacian). On the right of this main button is a number, which when negative, replace the normal second order non-linear term with a (dh/dx)**4. When the number takes the absolute value of 2 then other variants comes into play:


Apart from the variation in possible equations, the applet allow you to explore system size dependence, and whether boundaries are free (-1), periodic (0) or tilted to various degrees (>0).

Finally the applet allow you to play with artistic design of output, with regulating Color coding (numbers 1 to 7), and what criterium colors are assigned to interface (none(+-1), height(+-2), gradient(+-3), or growth speed(+-4), -sign switches background to black).

Finally you can adjust the drawing frequency and the speed of the simulations. Good parameters for slow brning simulation is (system size=300,color=1,criterium=3,line density=2)

Philosophy

Kuramoto-Sivashisky equation is the simplest model for a spatially unstable system which, due to non-linear couplings, reaches a steady state dynamics where no parts is ahead all the time. The equation describes how the front or interface h(x) develops in time. x is here the spatial extent, and h the height.

The key idea in this equation is that if h(x) is ahead of its immediate neighbors h(x-1) and h(x+1), then h(x) advances faster. This in itself is described by the equation h(after)-h(before) = (h(x)-h(x-1)) + (h(x)-h(x+1)) or in formal equation form dh/dt=-Laplace(h)=-dd(h)/ddx.

This instability is then ultimately remedied by corrections associated to additional growth of h at positions where some neighbors are very far ahead: dh/dt = -dd(h)/ddx - dddd(h)/ddddx + (dh/dx)(dh/dx)

In the applet we explore this equation as well as a number of variants associated to higher order instability terms, as well as higher order linear and non linear stabilization terms. These particular variants are regulated by the number just right on the to Kuramoto "botton" on the applet.

The large scale properties of the K-S. equation is similar to the KPZ equation, see Yakhot (1981) and Sneppen et al (1992). Probably this is true for most of the variants also.

Litterature:

Y. Kuramoto, T. Tsuzuki, "Persistent propagation of concentration waves in dissipative media far from thermal equilibrium"
Progr. Theoret. Phys. , 55 (1976) pp. 356369

Sivashinsky, G. I. (1983) "Instabilities, Pattern Formation and Turbulence in Flames"
Ann. Rev. Fluid Mech. 1B, 179.

V. Yakhot: "Large-scale properties of unstable systems governed by the Kuramoto-Sivashinksi equation"
Phys. Rev. A 24, 642 - 644 (1981)

K.Sneppen, J.Krug, MHJensen, C.Jayaprakash, and T.Bohr. "Dynamic scaling and crossover analysis for the Kuramoto - Sivashinsky equation."
Phys. Rev. A 46, R7351 - R7354 (1992).

T. Sams, K. Sneppen, M.H. Jensen, C. Ellegaard, B.E.Christensen and U. Trane. "Morphological instability in a growing yeast colony"
Physical Reviev Letters 79, 313 (1997).