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,^[1] who along the Daniel M. Michelson, presented the numerical solutions of the equation in the same year.^[2] The equation for ${\displaystyle u(\mathbf {x} ,t)}$ (where ${\displaystyle \mathbf {x} =(x,y)}$), which describes the perturbation amplitude of the planar flame front, reads as^[3]

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

where ${\displaystyle \nu }$ is a constant. Incorporating also the Rayleigh–Taylor instability of the flame, the equation modifies to

${\displaystyle {\frac {\partial u}{\partial t}}+{\frac {1}{2}}(\nabla u)^{2}-\nu \nabla ^{2}u-{\frac {1}{8\pi ^{2}}}\int |\mathbf {k} |e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')}u(\mathbf {x}$ ;,t)d\mathbf {k} d\mathbf {x} '+\gamma \left(u-\langle u\rangle \right)=0,\quad }

where ${\displaystyle \langle u\rangle (t)}$ is the spatially-averaged quantity of ${\displaystyle u}$, which is a time-dependent function and ${\displaystyle \gamma }$ is another constant.

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.^[4][5][6] Consider the 1d equation

${\displaystyle u_{t}+uu_{x}-\nu u_{xx}=\int _{-\infty }^{+\infty }e^{ikx}{\hat {u}}(k,t)dk,}$

where ${\displaystyle {\hat {u}}}$ is the Fourier transform of ${\displaystyle u}$. This has a solution of the form^[4][7]

${\displaystyle {\begin{aligned}u(x,t)&=-2\nu \sum _{n=1}^{2N}{\frac {1}{x-z_{n}(t)}},\\{\frac {dz_{n}}{dt}}&=-2\nu \sum _{l=1,l\neq n}^{2N}{\frac {1}{z_{n}-z_{l}}}-i\mathrm {sgn} (\mathrm {Im} z_{n}),\end{aligned}}}$

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

${\displaystyle {\begin{aligned}u(x,t)&=-\nu \sum _{n=1}^{2\pi }\cot {\frac {x-z_{n}(t)}{2}},\\{\frac {dz_{n}}{dt}}&=-\nu \sum _{l\neq n}\cot {\frac {z_{n}-z_{l}}{2}}-i\mathrm {sgn} (\mathrm {Im} z_{n})\end{aligned}}}$

These poles are interesting because in physical space, they correspond to locations of the cusps forming in the flame front.^[8]

• Kuramoto–Sivashinsky equation