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 Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously 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.

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