  We consider a one-dimensional Swift-Hohenberg equation coupled to a
conservation law, where both equations contain additional dispersive terms
breaking the reflection symmetry $x \mapsto -x$. This system exhibits a Turing
instability and we study the dynamics close to the onset of this instability.
First, we show that periodic traveling waves bifurcate from a homogeneous
ground state. Second, fixing the bifurcation parameter close to the onset of
instability, we construct modulating traveling fronts, which capture the
process of pattern-formation by modeling the transition from the homogeneous
ground state to the periodic traveling wave through an invading front. The
existence proof is based on center manifold reduction to a finite-dimensional
system. Here, the dimension of the center manifold depends on the relation
between the spreading speed of the invading modulating front and the linear
group velocities of the system. Due to the broken reflection symmetry, the
coefficients in the reduced equation are genuinely complex. Therefore, the main
challenge is the construction of persistent heteroclinic connections on the
center manifold, which correspond to modulating traveling fronts in the full
system.
