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In mathematics, the 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

$${\displaystyle u_{t}+\nabla ^{4}u+\nabla ^{2}u+{\frac {1}{2}}|\nabla u|^{2}=0,}$$

where $$\nabla ^{2}$$ is the Laplace operator and its square, $$\nabla ^{4}$$ is the biharmonic operator. The Kuramoto–Sivashinsky equation is known for its chaotic behavior.

Clarke's equation

References

Weisstein, Eric W. "Kuramoto-Sivashinsky Equation". MathWorld. Wolfram Research.
Kuramoto, Y. (1978). Diffusion-induced chaos in reaction systems. Progress of Theoretical Physics Supplement, 64, 346-367.
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).

Sivashinsky, G. I. (1980). "On flame propagation under conditions of stoichiometry". SIAM Journal on Applied Mathematics. 39 (1): 67–82. doi:10.1137/0139007.

(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

$${\displaystyle u_{t}+\nabla ^{4}u+\nabla ^{2}u+{\frac {1}{2}}|\nabla u|^{2}=0,}$$

where $$\nabla ^{2}$$ is the Laplace operator and its square, $$\nabla ^{4}$$ is the biharmonic operator. The Kuramoto–Sivashinsky equation is known for its chaotic behavior.

Clarke's equation

References

Weisstein, Eric W. "Kuramoto-Sivashinsky Equation". MathWorld. Wolfram Research.
Kuramoto, Y. (1978). Diffusion-induced chaos in reaction systems. Progress of Theoretical Physics Supplement, 64, 346-367.
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).

Sivashinsky, G. I. (1980). "On flame propagation under conditions of stoichiometry". SIAM Journal on Applied Mathematics. 39 (1): 67–82. doi:10.1137/0139007.

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