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 MeixnerPollaczek recurrence; accurate steady crest shapes ensue for N3. Squeezed crests ruled by a discretized Burgers equation ...

In this work, we study the one-dimensional stabilized Kuramoto Sivashinsky equation with additive uncorrelated stochastic noise. The Eckhaus stable band of the deterministic equation collapses to a narrow region near the center of the band. This is consistent with the behavior of the phase diffusion constants of these states. Some connections to the phenomenon of state selection in driven out o...

We prove a global well-posedness and regularity result of strong solutions to slightly modified Michelson-Sivashinsky equation in any spatial dimension the absence physical boundaries. Local-in-time (and regularity) space $W^{1,\infty}(\mathbb{R}^d)$ is established shown be if addition initial data either periodic or vanishes at infinity. The proof latter utilizes ideas previously introduced by...

We report the first observations of numerical "hopping" cellular flame patterns found in computer simulations of the Kuramoto-Sivashinsky equation. Hopping states are characterized by nonuniform rotations of a ring of cells, in which individual cells make abrupt changes in their angular positions while they rotate around the ring. Until now, these states have been observed only in experiments b...