Hidden symmetry in a Kuramoto-Sivashinsky initial-boundary value problem

TL;DR

This paper uncovers hidden symmetries in the bifurcation structure of the Kuramoto-Sivashinsky equation with Dirichlet boundary conditions, revealing complex stability and solution coexistence phenomena.

Contribution

It introduces a novel approach using extended periodic problems and symmetry principles to analyze bifurcations in the Kuramoto-Sivashinsky equation with boundary conditions.

Findings

Identification of two supercritical pitchfork bifurcations.

Existence of multiple coexisting stable solutions.

Complex bifurcation sequences including Hopf bifurcations.

Abstract

We investigate the bifurcation structure of the Kuramoto-Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and $\Otwo$ symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary conditions and one that breaks a shift-reflect symmetry of the extended system. Using Lyapunov-Schmidt reduction, we show both to be supercritical. We extend the primary branches by means of numerical continuation, and show that they lose stability in pitchfork, transcritical or Hopf bifurcations. Tracking the corresponding secondary branches reveals an interval of the viscosity parameter in which up to four stable equilibria and time-periodic solutions coexist. Since the study of the extended problem is indispensible for the explanation of the bifurcation structure, the Kuramoto-Sivashinsky problem with Dirichlet boundary conditions provides an elegant manifestation of hidden symmetry.