Symbolic dynamics for Kuramoto-Sivashinsky PDE on the line --- a computer-assisted proof

TL;DR

This paper provides a computer-assisted proof demonstrating the existence of symbolic dynamics and infinitely many periodic orbits for the Kuramoto-Sivashinsky PDE under specific boundary conditions and parameters.

Contribution

It introduces a novel computer-assisted method to rigorously establish symbolic dynamics and infinite periodic orbits for a nonlinear PDE.

Findings

Existence of symbolic dynamics proven for the PDE.

Countably infinite periodic orbits identified.

Method applicable to similar nonlinear PDEs.

Abstract

The Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter $\nu=0.1212$ is considered. We give a computer-assisted proof the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods.