Anisotropic Bose-Einstein condensates and completely integrable dynamical systems

TL;DR

This paper derives and analyzes a completely integrable Ermakov system from a Gaussian variational approach to anisotropic Bose-Einstein condensates, providing exact solutions and insights into condensate collapse dynamics.

Contribution

It introduces a novel connection between anisotropic BEC dynamics and Ermakov systems, proving their complete integrability and enabling exact analysis of collapse phenomena.

Findings

The dynamical system is shown to be a completely integrable Ermakov system.

Exact solutions are obtained for the condensate dynamics.

Collapse behavior is analyzed in detail using the solutions.

Abstract

A Gaussian ansatz for the wave function of two-dimensional harmonically trapped anisotropic Bose-Einstein condensates is shown to lead, via a variational procedure, to a coupled system of two second-order, nonlinear ordinary differential equations. This dynamical system is shown to be in the general class of Ermakov systems. Complete integrability of the resulting Ermakov system is proven. Using the exact solution, collapse of the condensate is analyzed in detail. Time-dependence of the trapping potential is allowed.