Flame Wrinkles From the Zhdanov-Trubnikov Equation

The Zhdanov-Trubnikov equation describing wrinkled premixed flames is studied, using pole-decompositions as starting points. Its one-parameter (-1< c <1) nonlinearity generalizes the Michelson-Sivashinsky equation (c=0) to a stronger Darrieus-Landau instability. The shapes of steady flame crests (or periodic cells) are deduced from Laguerre (or Jacobi) polynomials when c = -1, which numerical resolutions confirm. Large wrinkles are analysed via a pole density: adapting results of Dunkl relates their shapes to the generating function of Meixner-Pollaczek polynomials, which numerical results confirm for 1<c<0 (reduced stabilization). Although locally ill-behaved if c>0 (over-stabilization) such analytical solutions can yield accurate flame shapes for 0< c <0.6. Open problems are invoked.