Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuram- oto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigor- ously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.