Invariant manifolds and attractors of a periodic boundary-value problem for the Kuramoto–Sivashinsky equation with allowance for dispersion

A periodic boundary-value problem for the dispersive Kuramoto–Sivashinsky equation is considered. The stability of homogeneous equilibria is examined and an analysis of local bifurcations with a change in stability is performed. This analysis is based on the methods of the theory of dynamical systems with an infinite-dimensional space of initial conditions. Sufficient conditions for the presence or absence of invariant manifolds are found. Asymptotic formulas for some solutions are obtained.