Periodic traveling waves of the Kuramoto–Sivashinsky equation

A periodic boundary-value problem for the Kuramoto–Sivashinsky equation is considered. We prove that there exists a two-parameter family of traveling-wave solutions and obtain asymptotic formulas for them. We also prove that the set of such solutions forms a two-dimensional invariant manifold, which is a local attractor. The indicated solutions have different periods in the variable $t$, are unstable in the Lyapunov sense, but are stable in the Perron, Poincaré, and Zhukovsky senses. The study is based on the theory of invariant manifolds and Poincaré–Dulac normal forms.