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

TL;DR

This paper combines numerical methods and computer-assisted proofs to establish the existence of periodic orbits in the Kuramoto-Sivashinsky equation, advancing the rigorous verification of complex dynamical behaviors.

Contribution

It introduces a methodology for proving periodic orbits using functional zeros, Newton's algorithm, and interval arithmetic, applied specifically to the Kuramoto-Sivashinsky equation.

Findings

Numerical computation of periodic orbits using Newton's method.

Rigorous verification of orbit existence via interval arithmetic.

Development of a methodology applicable to various orbit types.

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 [19] for several types of orbits. An independent implementation, covering overlapping but different ground, using different functional setups appears in [33].