  We consider a dc-driven damped sine-Gordon model with a small nonlinear
spatial-disorder term, onto which a sinusoidal modulation is superimposed. It
describes, e.g., a weakly disordered system with a regular grain structure. We
demonstrate that, at the second order of the perturbation theory (with respect
to the weak spatial disorder), the periodically modulated disorder gives rise
to an effective periodic potential. Dynamics of a kink moving in this potential
is studied in the overdamped limit, using the adiabatic approximation, the main
objective being to consider depinning of a trapped kink. The analytical results
are compared with direct dynamical simulations of the underlying model, as well
as with numerical results using the collective-coordinate approach but without
the mean-field approximation. It is found that a critical force for the
depinning of a kink trapped by the periodically modulated weak spatial disorder
is much larger than that predicted by the mean-field approximation.
