Phase Transport beyond Local Pinning in the Kuramoto–Sivashinsky Equation

A homogeneous phase can be locally stable without capturing every solution of a spatially extended equation. We establish this distinction for the sine-forced Kuramoto–Sivashinsky equation on a circle of length 32π. At forcing amplitude A = 9/25, a computer-assisted proof constructs a real relative-periodic solution satisfying θ(x,t+T) = θ(x+16π,t) − 2π, with 50.36265677334806 < T < 50.36268699095121. Its mean phase transport is exactly −2π/T, while constant pinned equilibria are locally exponentially stable at the same amplitude. The existence proof uses a bounded Hilbert graph formulation, a 5,298-dimensional bordered Schur core, and a finite Neumann polynomial with controlled infinite-tail and floating-point errors. A separate full-space variational certificate proves local orbital exponential attraction in H², with asymptotic time and spatial phases. An analytic continuation argument establishes coexistence on an unspecified open interval of forcing amplitudes around 9/25. For each prescribed asymptotic exponential rate q ∈ (0.06324,1) per return, the amplitude neighbourhood may be smaller. No numerical interval width or quantitative basin radius is asserted. The results retain the explicit arithmetic assumptions stated in the paper. The accompanying files contain the preprint, its LaTeX source, and the source code, data, numerical certificates, and instructions for reproducing the computations.