= Kuramoto-Sivashinsky equation
{c}
{title2=$u_t+uu_x+u_{xx}+u_{xxxx}=0$}
= KS equation
{c}
{synonym}
The Kuramoto-Sivashinsky equation $u_t+uu_x+u_{xx}+u_{xxxx}=0$ combines destabilizing long-wave growth, stabilizing fourth-order <diffusion>, and nonlinear transport. Its potential form is $\Phi_t=-\Phi_{xx}-\Phi_{xxxx}+(\Phi_x)^2$, with $u=-2\Phi_x$. It arises as a long-wave phase <amplitude equation> near loss of the <Benjamin-Feir stability condition>. On a domain of period $P$, the zero solution's <Fourier modes> grow at rate $k^2-k^4$, $k=2\pi n/P$.
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