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, who, along with Daniel M. Michelson, presented numerical solutions of the equation in the same year. The evolution of deviations from planarity is described by an amplitude function u ( x , t ) {\displaystyle u(x,t)} . The 1D Michelson–Sivashinsky equation reads:
u t + 1 2 u x 2 = ν u x x + H ( u x ) , {\displaystyle u_{t}+{\frac {1}{2}}u_{x}^{2}=\nu u_{xx}+{\mathcal {H}}(u_{x}),}
where H {\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,
σ = | k | − ν k 2 . {\displaystyle \sigma =|k|-\nu k^{2}.}
For the variable v = u x {\displaystyle v=u_{x}} , the equation is given by
v t + v v x = ν v x x + H ( v x ) , {\displaystyle v_{t}+vv_{x}=\nu v_{xx}+{\mathcal {H}}(v_{x}),}
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
v t + v v x = ν v x x + H ( v x ) . {\displaystyle v_{t}+vv_{x}=\nu v_{xx}+{\mathcal {H}}(v_{x}).}
This has a solution of the form
v ( x , t ) = − 2 ν ∑ n = 1 2 N 1 x − z n ( t ) , d z n d t = − 2 ν ∑ l = 1 , l ≠ n 2 N 1 z n − z l − i s g n ( I m z n ) , {\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}}}
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