  Pattern-forming fronts are often controlled by an external stimulus which
progresses through a stable medium at a fixed speed, rendering it unstable in
its wake. By controlling the speed of excitation, such stimuli, or "triggers,"
can mediate pattern forming fronts which freely invade an unstable equilibrium
and control which pattern is selected. In this work, we analytically and
numerically study when the trigger perturbs an oscillatory pushed free front.
In such a situation, the resulting patterned front, which we call a pushed
trigger front, exhibits a variety of interesting phenomenon, including snaking,
non-monotonic wavenumber selection, and hysteresis. Assuming the existence of a
generic oscillatory pushed free front, we use heteroclinic bifurcation
techniques to prove the existence of trigger fronts in an abstract setting
motivated by the spatial dynamics approach. We then derive a leading order
expansion for the selected wavenumber in terms of the trigger speed.
Furthermore, we show that such a bifurcation curve is governed by the
difference of certain strong-stable and weakly-stable spatial eigenvalues
associated with the decay of the free pushed front. We also study prototypical
examples of these phenomena in the cubic-quintic complex Ginzburg Landau
equation and a modified Cahn-Hilliard equation.
