Kuramoto–Sivashinsky equation

In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation or flame equation) is a fourth-order nonlinear partial differential equation,[1] named after Yoshiki Kuramoto[2] and Gregory Sivashinsky, who derived the equation to model the diffusive instabilities in a laminar flame front in the late 1970s.[3][4] The equation reads as

${\displaystyle u_{t}+\Delta ^{2}u+\Delta u+{\frac {1}{2}}|\nabla u|^{2}=0,}$

A spatiotemporal plot of a simulation of the Kuramoto–Sivashinsky equation

where ${\displaystyle \Delta }$ is the Laplace operator and its square, ${\displaystyle \Delta ^{2}}$ is the biharmonic operator. The Kuramoto–Sivashinsky equation is known for its chaotic behavior.[5]

See also

• Clarke's equation

References

1. Weisstein, Eric W. "Kuramoto-Sivashinsky Equation". MathWorld. Wolfram Research.
2. Kuramoto, Y. (1978). Diffusion-induced chaos in reaction systems. Progress of Theoretical Physics Supplement, 64, 346-367.
3. Sivashinsky, G. S. (1977). Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations. In Dynamics of Curved Fronts (pp. 459-488).
4. Sivashinsky, G. I. (1980). "On flame propagation under conditions of stoichiometry". SIAM Journal on Applied Mathematics. 39 (1): 67–82. doi:10.1137/0139007.
5. Pathak, Jaideep; Hunt, Brian; Girvan, Michelle; Lu, Zhixin; Ott, Edward (2018). "Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach". Physical Review Letters. 120 (2): 024102. doi:10.1103/PhysRevLett.120.024102. ISSN 0031-9007.