Analytical and numerical aspects in solving the controlled 3D Gross-Pitaevskii equation
Renato Fedele, Dušan Jovanović, Sergio M. De Nicola +4 more
The results of recently developed investigations, that have been carried out within the framework of the controlling potential method (CPM), are reviewed. This method allows one to decompose a three dimensional (3D) Gross‐Pitaevskii equation (GPE) into the pair of coupled Schrödinger‐type equations. Under suitable mathematical conditions, the solutions of the 3D controlled GPE can be constructed from the solutions of a 2D linear Schrödinger equation (the transverse component of the GPE) coupled with a 1D nonlinear Schrödinger equation (the longitudinal component of the GPE). Such decomposition allows one to cast the solutions in the form of the product of the solutions of the transverse and the longitudinal components of the GPE. The coupling between these two equations is the functional of both the transverse and the longitudinal profiles. It is shown that the CPM can be used to obtain a new class of three‐dimensional solitary waves solutions of the GPE, which governs the dynamics of Bose‐Einstein condensates. By imposing an external controlling potential, the desired time‐dependent shape of the localized BECs is obtained. The stability of the exact solutions was checked with direct simulations of the time ‐dependent, three‐dimensional GPE. Our simulations show that the localized condensates are stable with respect to perturbed initial conditions.
AIP conference proceedings•Jan 1, 2009•0 citations