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

TL;DR

This paper develops an analytical method to construct localized solutions of the controlled 3D Gross-Pitaevskii equation, analyzing their stability and collapse behavior in Bose-Einstein condensates.

Contribution

It introduces a novel analytical approach that decomposes the 3D GPE into lower-dimensional equations, enabling explicit solution construction and stability analysis.

Findings

Identified conditions for stable localized solutions.

Derived criteria for collapse and non-collapse scenarios.

Provided exact analytical solutions for the controlled 3D GPE.

Abstract

On the basis of recent investigations, a newly developed analytical procedure is used for constructing a wide class of localized solutions of the controlled three-dimensional (3D) Gross-Pitaevskii equation (GPE) that governs the dynamics of Bose-Einstein condensates (BECs). The controlled 3D GPE is decomposed into a two-dimensional (2D) linear Schr\"{o}dinger equation and a one-dimensional (1D) nonlinear Schr\"{o}dinger equation, constrained by a variational condition for the controlling potential. Then, the above class of localized solutions are constructed as the product of the solutions of the transverse and longitudinal equations. On the basis of these exact 3D analytical solutions, a stability analysis is carried out, focusing our attention on the physical conditions for having collapsing or non-collapsing solutions.