Controlling potential traps for filtering solitons in Bose – Einstein condensates

We present a controlling potential method for solving the three-dimensional Gross – Pitaevskii equation (GPE), which governs the nonlinear dynamics of Bose – Einstein condensates (BECs) in an inhomogeneous potential trap. Our method allows one to construct ground and excited matter wave states whose longitudinal profiles can have bright solitons. This method provides the confining potential that filters and controls localized BECs. Moreover, it is predicted that, while the BEC longitudinal soliton profile is controlled and kept unchanged, the transverse profile may exhibit oscillatory breathers (unmatched case) or moving as a rigid body in the form of either coherent states (performing the Lissajous figures) or a Schrödinger cat state (matched case).