Michelson–Sivashinsky equation

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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.^[3][4][5][6] Consider the 1d equation

${\displaystyle v_{t}+vv_{x}=\nu v_{xx}+{\mathcal {H}}(v_{x}).}$

This has a solution of the form^[3][7]

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

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

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

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

In 1995,^[9] John W. Dold and Guy Joulin generalised the Michelson–Sivashinsky equation by introducing the second-order time derivative, which is consistent with the quadratic nature of the dispersion relation for the Darrieus–Landau instability. The Dold–Joulin equation is given by

${\displaystyle u_{tt}+{\mathcal {H}}\left(u_{t}-{\frac {1}{2}}u_{x}^{2}-\nu u_{xx}-{\mathcal {H}}(u_{x})\right)=0.}$

In 1992,^[10] Guy Joulin and Pierre Cambray extended the Michelson–Sivashinsky equation to include higher-order correction terms, following by an earlier incorrect attempt to derive such an equation by Gregory Sivashinsky and Paul Clavin.^[11] The Joulin–Cambray equation, in dimensional form, reads as

${\displaystyle u_{t}+{\frac {S_{L}}{2}}\left(1+{\frac {\epsilon }{2}}\right)u_{x}^{2}-\epsilon {\frac {S_{L}}{4}}\langle u_{x}^{2}\rangle ={\frac {S_{L}\epsilon }{2}}\left(1+{\frac {\epsilon }{2}}\right)\left[\nu \nabla ^{2}\phi +{\mathcal {H}}(u_{x})\right],}$

${\displaystyle \langle u\rangle (t)}$ denotes the spatial average of ${\displaystyle u}$, which is a time-dependent function.

Incorporating the Rayleigh–Taylor instability of the flame, one obtains the Rakib–Sivashinsky equation (named after Z. Rakib and Gregory Sivashinsky),^[12]

${\displaystyle u_{t}+{\frac {1}{2}}u_{x}^{2}=\nu u_{xx}+{\mathcal {H}}(u_{x})-\gamma \left(u-\langle u\rangle \right)=0,\quad }$

where ${\displaystyle \gamma }$ is another constant, related to gravity.

• Kuramoto–Sivashinsky equation