= Primary periodic bifurcation of the Kuramoto-Sivashinsky equation
Fix zero mean and period $P$ in the <Kuramoto-Sivashinsky equation>. At $P=2\pi$ the first <Fourier mode> becomes neutral and all higher modes are damped. For $k=2\pi/P$, expand $u=a\sin(kx)+b\sin(2kx)+\cdots$. The first equations are $\dot a=\lambda_1a+kab/2+\cdots$, $\dot b=\lambda_2b-ka^2/2+\cdots$, with $\lambda_j=(jk)^2-(jk)^4$. Slaving the second harmonic gives $\dot a=\lambda_1a+k^2a^3/(4\lambda_2)+\cdots$. Since $\lambda_2=-12$ at onset, this is a <supercritical bifurcation>. The small nonzero branch is stable in the sense of <orbital stability> against disturbances of the same fixed period, modulo <translation>; stability against disturbances of larger period does not follow from this local argument.
Back to article page
