for the fermionic part of the mixture. Equations (\[bose-fermi-eqns\]) were also derived in [@Karpiuk1; @Karpiuk2] by means of a Lagrangian density approach.

In the following we shall consider Eq. (\[bose-fermi-eqns\]) in a quasi one dimensional context by assuming the bosons and fermions confined in a cylindrical trap with transverse trapping frequency $\omega_{\perp}$ and negligible x-axial confinement. We also assume that the ground state energy of the transverse trapping potential $\hbar \omega_{\perp}$ is larger than the 3D ground state energies of the bosons and the fermions. Within this approximation the bosonic and fermionic wavefunctions can be factorized as $\psi=\psi(x)\psi_{\perp}(y,z)$ with the transverse wavefunction taken as the ground state of the trap potential independently of the longitudinal behavior and statistics [@das]. By integrating over transverse coordinates one obtains effective 1D equations which are formally identical to Eqs. (\[bose-fermi-eqns\]) but with boson-boson and boson-fermion interactions rescaled according to: $g_{b}\rightarrow 2 \hbar
\omega_{\perp} g_{b},\;\;\; g_{bf}\rightarrow 2 \hbar
\omega_{\perp} g_{bf}$.

Notice that the fact that fermionic and bosonic densities appear in the equations of bosons and fermions, respectively, suggests to consider them as effective potentials and solve the quasi-classical equation of motion by means of a self consistent procedure. Starting with a trial function for 