Scaling properties of the Kuramoto-Sivashinsky equation

The Kuramoto-Sivishinsky model describes the dynamics of a cellular flame front. It has been known for some time that on scales large compared with the size of a cell the front appears to be a self-affine fractal which has noisy dynamics in 1+1 dimensions. We use the inverse method of Lam and Sander (Phys. Rev. Lett. 71, 561 (1993)) to show explicitly how the scaling occurs and how deterministic chaos at small scales develops into noisy dynamics at large scales, and how a small scale pattern becomes a large scale disordered fractal via an intermediate scaling regime.