Sivashinsky (1977) tailored the first equation to describe the dynamics of unforced wrinkled fronts
of premixed gaseous flames when the fresh‐to‐burnt gas density jump is small: it accounts for local
curvature effects, an eikonal non‐linearity and the Darrieus‐Landau hydrodynamic – hence nonlocal –
instability. Locating the front‐slope complex poles in principle gives access to closed‐form shapes for
isolated front crests or for periodic patterns, and to the wrinkling induced increase in burning speed.
Various ways to effectively locate the poles and get the flame‐front profiles are presented: exact
resolutions, a near‐exact one fallen “out of the blue”, and asymptotically correct approaches that
require analytical resolutions of linear singular integral equations; and, of course, numerical checks.
It is next shown how to generalize all this to flames subject to some forcing, e.g. shear‐flows. Results
for steady fronts are presented, and current analyses of unsteady ones are sketched.
An equation generalizing Sivashinsky’s is finally introduced, and various analytical solutions thereof
are displayed. Surprisingly enough, some of those can also be useful to model steady curved flames
in wide channels when the density jump is not small; steady forcing could even be accounted for ...
once the (still ongoing‐) resolution of a suitable nonlocal Riemann‐Hilbert problem is completed.