  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
numerical simulations.
