(second and first curve from the top, respectively). The periodic potential, scaled by a factor 10 for graphical convenience, is also shown with a thin line at the bottom. Parameter values are the same as in Fig. \[fig3\] with $N_b=1$. [**(b)**]{} Time evolution of the bosonic and fermionic densities of the metastable IA gap-soliton obtained from the quasi-classical Eqs. (\[bose-fermi-eqns\]) on a line of length $L=40 \pi$. Panels [**(c)**]{}. Same as panel (a) but for the OS state obtained after the transition. Panel [**(d)**]{}. Time evolution of the wavefunction of the lowest energy state inside the condensate. Plotted quantities are in normalized units.[]{data-label="fig5"}](fig5d.eps "fig:"){width="4.2cm" height="4.6cm"}

We see that after a time $t\approx 60$ the IA state decays into the stable OS wavefunction depicted in Fig. \[fig5\]c. Notice that the IA-OS transition also induces changes in the fermionic states. In particular we see that the first excited state asymptotically decays into the Bloch state at the bottom of the fermionic band (see top curve of Fig.\[fig5\]c), while the lowest energy fermionic state remains well localized inside the gap soliton (see the time evolution reported in panel (c)). The IA-OS transition in Fig. \[fig5\]c occurs therefore with the expulsion of one fermion 