TY - JOUR

T1 - Mathematical and physical aspects of controlling the exact solutions of the 3D Gross–Pitaevskii equation

AU - Fedele, Renato

AU - Jovanovic, Dusan

AU - De Nicola, Sergio

AU - Eliasson, Bengt

AU - Shukla, Padma

PY - 2010/1/18

Y1 - 2010/1/18

N2 - The possibility of the decomposition of the three-dimensional (3D) Gross–Pitaevskii equation (GPE) into a pair 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.

AB - The possibility of the decomposition of the three-dimensional (3D) Gross–Pitaevskii equation (GPE) into a pair 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.

KW - Bose-Einstein condensates

KW - Gross-Pitaevskii equation

UR - http://www.scopus.com/inward/record.url?scp=73149104853&partnerID=8YFLogxK

UR - http://www.sciencedirect.com/science/article/pii/S0375960109015023

U2 - 10.1016/j.physleta.2009.11.069

DO - 10.1016/j.physleta.2009.11.069

M3 - Article

SN - 0217-7323

VL - 374

SP - 788

EP - 795

JO - Modern Physics Letters A

JF - Modern Physics Letters A

IS - 5

ER -