Exact solutions of the modified Gross-Pitaevskii equation in `smart' periodic potentials in the presence of external source

We report wide class of exact solutions of the modified Gross-Pitaevskii equation (GPE) in `smart’ Jacobi elliptic potentials: $V(\xi)=-V_{0}{\rm sn(\xi,m)}$,$V(\xi)=-V_{0}{\rm cn(\xi,m)}$,and$V(\xi)=-V_{0}{\rm dn(\xi,m)}$ in the presence of external source. Solitonlike solutions, singular solutions, and periodic solutions are found using a recently developed fractional transform: $\rho (\xi )=\frac{A+B{f}^2}{1+Df}$, where $f$ is the respective Jacobi elliptic function and the amplitude parameters $A$, $B$, and $D$ {\it nonzero}. These results generalize those contained in (Paul T, Richter K and Schlagheck P 2005 \emph{Phys. Rev. Lett.} {\bf 94}, 020404) for nonzero trapping potential.