Symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line II

TL;DR

This paper proves the existence of infinitely many homoclinic and heteroclinic orbits in the Kuramoto-Sivashinsky PDE on the line, using a computer-assisted method and a novel algorithm for rigorous integration of variational equations.

Contribution

It introduces a new algorithm for rigorous integration of variational equations in dissipative PDEs and applies it to establish complex orbit structures in the Kuramoto-Sivashinsky PDE.

Findings

Existence of infinitely many homoclinic orbits

Existence of heteroclinic orbits between periodic solutions

Development of a new computer-assisted rigorous integration method

Abstract

We prove the existence of infinite number of homoclinic and heteroclinic orbits to two periodic orbits for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and for some fixed parameter value of the system. The proof is computer assisted and it is based on a new algorithm for rigorous integration of the variational equation for a class of dissipative PDEs on the torus.