Michelson–Sivashinsky equation

In combustion, Michelson–Sivashinsky equation describes the evolution of a premixed flame front, subjected to the Darrieus–Landau instability, in the small heat release approximation. The equation was derived by Gregory Sivashinsky in 1977, who along the Daniel M. Michelson, presented the numerical solutions of the equation in the same year. Let the planar flame front, in a uitable frame of reference be on the $$xy$$-plane, then the evolution of this planar front is described by the amplitude function $$u(\mathbf x,t)$$ (where $$\mathbf x=(x,y)$$) describing the deviation from the planar shape. The Michelson–Sivashinsky equation, reads as


 * $$\frac{\partial u}{\partial t} + \frac{1}{2}(\nabla u)^2 - \nu \nabla^2 u - \frac{1}{8\pi^2} \int |\mathbf k| e^{i\mathbf k\cdot(\mathbf x-\mathbf x')}u (\mathbf x;,t) d\mathbf kd\mathbf x'=0,$$

where $$\nu$$ is a constant. Incorporating also the Rayleigh–Taylor instability of the flame, one obtains the Rakib–Sivashinsky equation (named after Z. Rakib and Gregory Sivashinsky),


 * $$\frac{\partial u}{\partial t} + \frac{1}{2}(\nabla u)^2 - \nu \nabla^2 u - \frac{1}{8\pi^2} \int |\mathbf k| e^{i\mathbf k\cdot(\mathbf x-\mathbf x')}u (\mathbf x;,t) d\mathbf kd\mathbf x' + \gamma \left(u - \langle

u \rangle \right)=0, \quad $$

where $$\langle u \rangle(t)$$ denotes the spatial average of $$u$$, which is a time-dependent function and $$\gamma$$ is another constant.

N-pole solution
The equations, in the absence of gravity, admits an explicit solution, which is called as the N-pole solution since the equation admits a pole decomposition,as shown by Olivier Thual, Uriel Frisch and Michel Hénon in 1988. Consider the 1d equation


 * $$u_t + u u_x - \nu u_{xx} = \int_{-\infty}^{+\infty} e^{ikx} \hat u(k,t) dk,$$

where $$\hat u$$ is the Fourier transform of $$u$$. This has a solution of the form


 * $$\begin{align}u(x,t) &= -2\nu \sum_{n=1}^{2N} \frac{1}{x-z_n(t)}, \\

\frac{dz_n}{dt} &= -2\nu \sum_{l=1,l\neq n}^{2N} \frac{1}{z_n-z_l} - i \mathrm{sgn}(\mathrm{Im} z_n), \end{align}$$

where $$z_n(t)$$ (which appear in complex conjugate pairs) are poles in the complex plane. In the case periodic solution with periodicity $$2\pi$$, the it is sufficient to consider poles whose real parts lie between the interval $$0$$ and $$2\pi$$. In this case, we have


 * $$\begin{align}

u(x,t) &= -\nu \sum_{n=1}^{2\pi} \cot\frac{x-z_n(t)}{2}, \\ \frac{dz_n}{dt} &= -\nu \sum_{l\neq n} \cot\frac{z_n-z_l}{2} - i \mathrm{sgn}(\mathrm{Im} z_n) \end{align}$$

These poles are interesting because in physical space, they correspond to locations of the cusps forming in the flame front.