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

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

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

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