boson-boson interactions. The number of fermions trapped in a BFM gap soliton and the energies of the bound states are also calculated both numerically and analytically. Finally, in the last section the main results of the paper are briefly discussed and summarized.

Equations of motion for a Bose-Fermi mixture
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To derive equations of motion of a BFM it is convenient to start from the second quantized Hamiltonian of a set of bosons and fermions in interaction $$\begin{aligned}
&&\hat H =\int dr \hat\psi_b^\dagger(r)\hat H_b^0 \hat\psi_b
(r)+\int dr \hat\psi_f^\dagger(r)\hat H_f^0 \hat\psi_f (r)+
\nonumber \\ && \frac12 \int\int dr
dr'\hat\psi_b^\dagger(r)\hat\psi_b^\dagger(r')U_{bb}(r-r')
\hat\psi_b(r')\hat\psi_b(r)+ \nonumber \\&& \int\int dr
dr'\hat\psi_b^\dagger(r)\hat\psi_f^\dagger(r')U_{bf}(r-r')
\hat\psi_b(r')\hat\psi_f(r),\end{aligned}$$ where $\hat H_b^0 = -\frac{\hbar^2}{2 m_b}\nabla^2+ V_{ext}^b,
\;\hat H_f^0=-\frac{\hbar^2}{2 m_f}\nabla^2+ V_{ext}^f$, are single particle hamiltonians accounting for the kinetic energy and for the external potential confining the bosons and the fermions, respectively, while $\hat\psi_b^\dagger(r), \hat\psi_b(r)$, $\hat\psi_f^\dagger(r), \hat\psi_f(r)$ are usual bosonic and fermionic field operators satisfying $[\hat\psi_b(r),\hat\psi_b(r')]=
[\hat\psi_b^\dagger(r),\hat\psi_b^\dagger(r')]=0,
\;[\hat\psi_b(r),\hat\psi_b^\dagger(r')]=\delta(r-r')$ (and similar relations for the fermionic fields with commutator replaced by the anti-commutator). Notice that the fermions are assumed spin polarized and therefore non interacting, so that only the boson-boson and boson-fermion interactions appear in the Hamiltonian. Using the above operator algebra, the Heisenberg equation of motion 