Two–dimensional stability of the “pole” solutions of the sivashinsky equation

The two–dimensional stability of the exact solution of the Sivashinsky equation governing the evolution of a curved flame surface in the hydrodynamic approximation is studied. It is shown that the one–dimensional pole solution of this equation governing a local minimum of the surface is stable with respect to small two–dimensional perturbations. The problem is solved under the assumption that the perturbations are small at a distance from the local minimum. Stable one–dimensional solutions may be used to verify numerical simulation of the surface of a hydrodynamically unstable flame and also to construct two–dimensional solutions of the Sivashinsky equation.