Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0908.1151 · The European Physical Journal B

Stability and collapse of localized solutions of the controlled three-dimensional Gross-Pitaevskii equation

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

Published Feb 16, 2010• 3 citations
arXivHTMLPDF
DOI
10.1140/epjb/e2010-00052-3

Related papers

Search for an Analytical Solution in the Three-Dimensional Gross–Pitaevskii Equation

В. С. Лапонин, Nadezhda Petrovna Savenkova

Computational Mathematics and Modeling•Mar 20, 2017•0 citations

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

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

Renato Fedele, Dušan Jovanović, Sergio M. De Nicola +2 more

Physics Letters A•Dec 3, 2009•7 citations

Soliton solutions of the 3D Gross-Pitaevskii equation by a potential control method

Renato Fedele, Bengt Eliasson, Fernando Haas +5 more

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.

arXiv:0907.1595•AIP conference proceedings•Jan 1, 2010•4 citations

Stability analysis and continuation for the coupled Gross–Pitaevskii equations

Sirilak Sriburadet, Yin-Tzer Shih, C.‐S. Chien

Computers & Mathematics with Applications•Mar 13, 2019•3 citations

Three-dimensional simulation of jet formation in collapsing condensates

Weizhu Bao, D Jaksch, Peter A. Markowich

We numerically study the behaviour of collapsing and exploding condensates using the parameters of the experiments by Donley et al (2001 Nature 412 295). Our studies are based on a full three-dimensional numerical solution of the Gross–Pitaevskii equation (GPE) including three-body loss. We determine the three-body loss rate from the number of remnant condensate atoms and collapse times, and obtain only one possible value which fits with the experimental results. We then study the formation of jet atoms by interrupting the collapse, and find very good agreement with the experiment. Furthermore, we investigate the sensitivity of the jets to the initial conditions. According to our analysis, the dynamics of the burst atoms is not described by the GPE with three-body loss incorporated.

arXiv:cond-mat/0307344•Journal of Physics B Atomic Molecular and Optical Physics•Dec 17, 2003•50 citations
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.
We numerically study the behaviour of collapsing and exploding condensates using the parameters of the experiments by Donley et al (2001 Nature 412 295). Our studies are based on a full three-dimensional numerical solution of the Gross–Pitaevskii equation (GPE) including three-body loss. We determine the three-body loss rate from the number of remnant condensate atoms and collapse times, and obtain only one possible value which fits with the experimental results. We then study the formation of jet atoms by interrupting the collapse, and find very good agreement with the experiment. Furthermore, we investigate the sensitivity of the jets to the initial conditions. According to our analysis, the dynamics of the burst atoms is not described by the GPE with three-body loss incorporated.