Kuramoto-Sivashinsky PDE

Spatiotemporal chaos in 1D — flame fronts, reaction-diffusion, turbulence

∂ₜu + u·∂ₓu + ∂ₓₓu + ∂ₓₓₓₓu = 0

Domain size L L = 80
Viscosity ν ν = 1.00
Speed x4
Colormap Fire
The KS equation models unstable flame fronts and thin film dynamics. It has negative viscosity (∂ₓₓu drives instability) balanced by hyperviscosity (∂ₓₓₓₓu). The result: deterministic chaos with a positive Lyapunov exponent. The space-time plot above shows the characteristic turbulent pattern.