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Physics Letters A

Mathematical and physical aspects of controlling the exact solutions of the 3D Gross–Pitaevskii equation

Renato Fedele, Dušan Jovanović, Sergio M. De Nicola, Bengt Eliasson, Padma Kant Shukla

Published Dec 3, 2009• 7 citations
Publisher
DOI
10.1016/j.physleta.2009.11.069

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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.

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arXiv:0907.1595•AIP conference proceedings•Jan 1, 2010•4 citations
The possibility of the decomposition of the three-dimensional (3D) Gross–Pitaevskii equation (GPE) into apair of coupled Schrödinger-type equations, is investigated. It is shown that, under suitable mathematical conditions, it is possible to construct the exact controlled solutions of the 3D GPE from the solutions of a linear 2D Schrödinger equation coupled with a 1D nonlinear Schrödinger equation (the transverse and longitudinal components of the GPE, respectively). The coupling between these two equations is the functional of the transverse and the longitudinal profiles. The applied method of nonlinear decomposition, called the controlling potential method (CPM), yields the full 3D solution in the form of the product of the solutions of the transverse and longitudinal components of the GPE. It is shown that the CPM constitutes a variational principle and sets up a condition on the controlling potential well. Its physical interpretation is given in terms of the minimization of the (energy) effects introduced by the control. The method is applied to the case of a parabolic external potential to construct analytically an exact BEC state in the form of a bright soliton, for which the quantitative comparison between the external and controlling potentials is presented.
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.
We present a class of three‐dimensional solitary waves solutions of the Gross‐Pitaevskii (GP) equation, which governs the dynamics of Bose‐Einstein condensates (BECs). By imposing an external controlling potential, a desired time‐dependent shape of the localized BEC excitation is obtained. The stability of some obtained localized solutions is checked by solving the time‐dependent GP equation numerically with analytic solutions as initial conditions. The analytic solutions can be used to design external potentials to control the localized BECs in experiment.