When the steady states at infinity become unstable through a
pattern forming bifurcation, a travelling wave may bifurcate
into a modulated front which is time-periodic in a moving
frame. This scenario has been studied by B. Sandstede and A. Scheel
for a class of reaction-diffusion systems on the real line.
Under general assumptions, they showed that the modulated
fronts exist and are spectrally stable near the bifurcation
point. Here we consider a model problem for which
we can prove the nonlinear stability of these solutions with
respect to small localized perturbations.
This result does not follow from the spectral stability, because
the linearized operator around the modulated front has essential
spectrum up to the imaginary axis. The analysis is illustrated by