Numerical Computations and Computer Assisted Proofs of Periodic Orbits of the Kuramoto--Sivashinsky Equation

Abstract

We present numerical results and computer assisted proofs of the existence of periodic orbits for the Kuramoto--Sivashinky equation. These two results are based on writing down the existence of periodic orbits as zeros of functionals. This leads to the use of Newton's algorithm for the numerical computation of the solutions and, with some a posteriori analysis in combination with rigorous interval arithmetic, to the rigorous verification of the existence of solutions. This is a particular case of the methodology developed in [J.-L. Figueras, J.-P. Lessard, M. Gameiro, and R. de la Llave, preprint, arXiv:1605.01086, 2016] for several types of orbits. An independent implementation, covering overlapping but different ground, using different functional setups, appears in [J.-P. Lessard and M. Gameiro, A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto-Sivashinsky PDE, manuscript].