Michelson–Sivashinsky equation

In combustion, the 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 with Daniel M. Michelson, presented numerical solutions of the equation in the same year.^[2] The evolution of deviations from planarity is described by an amplitude function ${\displaystyle u(x,t)}$. The 1D Michelson–Sivashinsky equation reads:

${\displaystyle u_{t}+{\frac {1}{2}}u_{x}^{2}=\nu u_{xx}+{\mathcal {H}}(u_{x}),}$

where ${\displaystyle {\mathcal {H}}}$ is the Hilbert transform. This is essentially the Burgers' equation with an additional non-local integral term. The Michelson–Sivashinsky equation represents the Darrieus–Landau instability, dictated by the dispersion relation, close to the instability onset,

${\displaystyle \sigma =|k|-\nu k^{2}.}$

For the variable ${\displaystyle v=u_{x}}$, the equation is given by

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

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

${\displaystyle v_{t}+vv_{x}=\nu v_{xx}+\int _{-\infty }^{+\infty }|k|e^{ikx}{\hat {v}}(k,t)dk,}$

where ${\displaystyle {\hat {v}}}$ is the Fourier transform of ${\displaystyle v}$. This has a solution of the form^[3][7]

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

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

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

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

${\displaystyle {\frac {\partial ^{2}\varphi }{\partial t^{2}}}+{\mathcal {I}}\left({\frac {\partial \varphi }{\partial t}}-{\frac {1}{2}}(\nabla \varphi )^{2}-{\mathcal {M}}_{t}\nabla ^{2}\varphi -{\mathcal {I}}(\varphi )\right)=0,}$

where ${\displaystyle {\mathcal {I}}(e^{i\mathbf {k} \cdot \mathbf {x} })=|\mathbf {k} |e^{i\mathbf {k} \cdot \mathbf {x} }}$ corresponds to the non-local integral operator.

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

${\displaystyle {\frac {\partial \phi }{\partial t}}+{\frac {S_{L}}{2}}\left(1+{\frac {\epsilon }{2}}\right)|\nabla \phi |^{2}+\epsilon {\frac {S_{L}}{4}}\langle |\nabla \phi |^{2}\rangle ={\frac {S_{L}\epsilon }{2}}\left(1+{\frac {\epsilon }{2}}\right)\left({\mathcal {M}}_{t}\nabla ^{2}\phi +I(\phi ,\mathbf {x} )\right).}$

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

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

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

• Kuramoto–Sivashinsky equation