Bifurcations of spatially inhomogeneous solutions of a boundary value problem for the generalized Kuramoto–Syvashinsky equation

In this paper, a differential partial equation with an unknown function of three variables time and two spatial variables – is considered. The given equation is commonly called the generalized Kuramoto–Sivashinsky (gKS) equation. This equation represents a model of the formation of a nanorelief on a surface by ion bombardment. In the work, this equation is considered with the homogeneous Neumann boundary conditions. Local bifurcations of spatially inhomogeneous equilibrium states is studied in the case of their stability changes. It is shown that the inhomogeneous surface relief can occur when the stability of the homogeneous states of equilibrium changes. The conditions were obtained for coefficients when the stability changes. In the cases close to critical cases the local bifurcation problems are considered. It was shown that a question about the formation of inhomogeneous surface relief from a mathematical point of view is reduced to the study of auxiliary ordinary differential equations which are called a Poincare–Dulac normal form. The stability analysis of spatially homogeneous equilibrium states is given, as well as local bifurcations are studied in the case of their stability changes. The method of invariant manifolds coupled with the normal form theory were used to solve this problem. For the bifurcating solutions the asymptotic formulas are given.