Lattice dark solitons in the linear potential

We deal with the nonlinear lattice
$$i\dot {\psi }_n+(1-|\psi _n|^2)( \psi _{n+1}+ \psi _{n-1}-2\psi _n)+2(\rho ^2-|\psi _n|^2)\psi _n+\gamma n\psi _n=0$$
subject to the finite density boundary conditions. It is shown that it is integrable by means of the inverse scattering technique. Soliton solutions undergo periodic motion with the frequency $\gamma$. Small amplitude limit of the model is discussed.