bound states trapped inside attractive gap solitons are also derived and the results compared with those obtained from direct numerical self-consistent calculations. We find that the filling factor of a gap soliton follows the law $n_f\propto \sqrt{|\chi_{bf}|N_b}$, where $N_b$ is the number of bosons, $n_f$ the number of fermions trapped in the condensate and $\chi_{bf}$ the strength of the boson-fermion interaction. The possibility to change $\chi_{bf}$ in a wide interval of values using Feshbach resonances suggests the possibility of an experimental check of this results. In particular we expect that, as the boson-fermion interaction is increased, the progressive filling of the gap soliton should give rise to a depletion of fermions in the OL which follows the above law and which could be monitored by imaging techniques.

The paper is organized as follow. In section II we introduce the model and derive quasi-classical equation of motion of Bose-Fermi mixtures directly from a second quantized formulation of problem. In section III we investigate the energy spectrum for BFM in attractive and repulsive parabolic potentials and discuss quantum dots (antidots) analogies. In section IV we consider BFM in optical lattices and show the existence of gap solitons both for repulsive and attractive 