A posteriori verification of invariant objects of evolution equations: periodic orbits in the Kuramoto-Sivashinsky PDE

In this paper, a method to compute periodic orbits of the Kuramoto-Sivashinsky PDE via rigorous numerics is presented. This is an application and an implementation of the theoretical method introduced in [1]. Using a Newton-Kantorovich type argument (the radii polynomial approach), existence of solutions is obtained in a weighted l∞ Banach space of Fourier coefficients. Once a proof of a periodic orbit is done, an associated eigenvalue problem is solved and Floquet exponents are rigorously computed, yielding proofs that some periodic orbits are unstable. Finally, a predictor- corrector continuation method is introduced to rigorously compute global smooth branches of periodic orbits. An alternative approach and independent implementation of [1] appears in [2].