  We consider the one-dimensional Swift-Hohenberg equation coupled to a
conservation law. As a parameter increases the system undergoes a Turing
bifurcation. We study the dynamics near this bifurcation. First, we show that
stationary, periodic solutions bifurcate from a homogeneous ground state.
Second, we construct modulating traveling fronts which model an invasion of the
unstable ground state by the periodic solutions. This provides a mechanism of
pattern formation for the studied system. The existence proof uses center
manifold theory for a reduction to a finite-dimensional problem. This is
possible despite the presence of infinitely many imaginary eigenvalues for
vanishing bifurcation parameter since the eigenvalues leave the imaginary axis
with different velocities if the parameter increases. Furthermore, compared to
non-conservative systems, we address new difficulties arising from an
additional neutral mode at Fourier wave number $k=0$ by exploiting that the
amplitude of the conserved variable is small compared to the other variables.
