from direct numerical integrations of Eqs. (\[bose-fermi-eqns\]) on a line of length $L=40 \pi$. Space and time are in normalized units.[]{data-label="fig4"}](fig4b.eps "fig:"){width="4.3cm" height="4.8cm"}

To this regard we consider BFM with repulsive bosonic-bosonic $\chi_{b}>0$ and attractive bosonic-fermionic $\chi_{bf}<0$ interactions. In Fig. \[fig3\]a the bosonic and fermionic densities corresponding to an onsite symmetric (OS) gap soliton, are depicted (notice that the soliton is symmetric around a minimum of the potential). We see that while the bosonic density is localized in almost a single potential well, the fermionic density is very extended in space and presents a hump in correspondence of the BEC. In Fig. \[fig3\]b the effective potentials, the sketch of the band structure and the energy levels of localized states for both bosons and fermions are shown. Notice that the energy of the OS gap soliton lies in the gap between the first two bands as expected for repulsive bosonic interactions, while the fermionic level responsible for the density hump in Fig. \[fig3\]a, lies below the lowest energy band at the bottom of the Fermi sea. As for the parabolic case, gap soliton states with trapped fermionic levels inside can be seen a matter wave realization of quantum dots.

More complicated 