Bifurcations of Spatially Inhomogeneous Solutions of a Boundary Value Problem for the Generalized Kuramoto–Sivashinsky Equation

Abstract

In this paper, the generalized Kuramoto–Sivashinsky (KS) equation with homogeneous Neumann boundary conditions is considered. The KS equation describes the formation of nano-scale patterns on a surface under ion beam sputtering. It is shown that the inhomogeneous surface relief structures can occur when there is an exchange of stabilities of the equilibrium points. Stability analysis of spatially homogeneous equilibrium states is given, as well as local bifurcations are studied in the case their stability changes. The method of invariant manifolds coupled with the normal form theory has been used to solve this problem. For the bifurcating solutions the asymptotic formulas are found.