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We consider the one-dimensional Swift-Hohenberg equation coupled to aconservation law. As a parameter increases the system undergoes a Turingbifurcation. We study the dynamics near this bifurcation. First, we show thatstationary, periodic solutions bifurcate from a homogeneous ground state.Second, we construct modulating traveling fronts which model an invasion of theunstable ground state by the periodic solutions. This provides a mechanism ofpattern formation for the studied system. The existence proof uses centermanifold theory for a reduction to a finite-dimensional problem. This ispossible despite the presence of infinitely many imaginary eigenvalues forvanishing bifurcation parameter since the eigenvalues leave the imaginary axiswith different velocities if the parameter increases. Furthermore, compared tonon-conservative systems, we address new difficulties arising from anadditional neutral mode at Fourier wave number $k=0$ by exploiting that theamplitude of the conserved variable is small compared to the other variables.

Modulating traveling fronts for the Swift-Hohenberg equation in the case of an additional conservation law