  We investigate the nonlinear dynamics of the Peyrard-Bishop DNA model taking
into account site dependent inhomogeneities. By means of the multiple-scale
expansion in the semi-discrete approximation, the dynamics is governed by the
perturbed nonlinear Schrodinger equation. We carry out a multiple-scale soliton
perturbation analysis to find the effects of the variety of nonlinear
inhomogeneities on the breatherlike soliton solution. During the crossing of
the inhomogeneities, the coherent structure of the soliton is found stable. The
global shape of the inhomogeneous molecule is merged with the shape of the
homogeneous molecule. However, the velocity, the wavenumber and the angular
frequency undergo a time-dependent correction that is proportional to initial
width of the soliton and depends on the nature of the inhomogeneities.
