  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.
