Exact solutions of the Gross-Pitaevskii equation in periodic potential in the presence of external source

Exact periodic solutions, solitonlike solutions, singular solitary, and singular trigonometric wave solutions of the time-dependent Gross-Pitaevskii equation (GPE) with elliptic function potential in the presence of external source are analyzed. A simple approach that applies equally to both attractive and repulsive time-dependent GPE and allows one to find an extensive list of explicit periodic solutions of the GPE in terms of the Jacobian elliptic functions is developed. In the limit as the elliptic modulus tends to unity or to zero, the linear solutions, in either the Jacobian elliptic cosine or the Jacobian elliptic function of third order, give solitonlike solutions, while the rational solutions in these elliptic functions lead to singular solitary or trigonometric wave solutions. The stability of these solutions is investigated numerically.