Premixed-flame shapes and polynomials

The nonlinear nonlocal Michelson-Sivashinsky equation for isolated crests of unstable flames is studied, using pole-decompositions as starting point. Polynomials encoding the numerically computed 2N flame-slope poles, and auxiliary ones, are found to closely follow a Meixner-Pollaczek recurrence; accurate steady crest shapes ensue for N≥3. Squeezed crests ruled by a discretized Burgers equation involve the same polynomials. Such explicit approximate shapes still lack for finite-N pole-decomposed periodic flames, despite another empirical recurrence.