Stability of matter–wave soliton in a time-dependent complicated trap

We examine the possibility to generate localized structures in effectively one-dimensional Gross–Pitaevskii with a time-dependent scattering length and a complicated potential. Through analytical methods invoking a generalized lens-type transformation and the Darboux transformation, we present the integrable condition for the Gross–Pitaevskii equation and obtain the exact analytical solution which describes the modulational instability and the propagation of bright solitary waves on a continuous wave background. The dynamics and stability of this solution are analyzed. Moreover, by employing the extended tanh-function method we obtain the exact analytical solutions which describes the propagation of dark and other families of solitary waves.

► We obtain analytically some interesting soliton solutions in BECs.
► We generate bright solitons for BECs with complex potential.
► We then study their stabilities.
► The model may have wider applications.