Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. I. General concept

TL;DR

This paper introduces a variational approach using coupled Gaussian functions to model Bose-Einstein condensates with long-range interactions, deriving dynamical equations and analyzing stability to explore new phenomena.

Contribution

The paper develops a novel variational method with coupled Gaussian functions for BECs with long-range interactions, deriving analytical dynamical equations from the Gross-Pitaevskii equation.

Findings

Derived dynamical equations for variational parameters.

Analyzed stability of solutions using nonlinear dynamics.

Set the stage for exploring new phenomena in BECs with long-range interactions.

Abstract

The variational method of coupled Gaussian functions is applied to Bose-Einstein condensates with long-range interactions. The time-dependence of the condensate is described by dynamical equations for the variational parameters. We present the method and analytically derive the dynamical equations from the time-dependent Gross-Pitaevskii equation. The stability of the solutions is investigated using methods of nonlinear dynamics. The concept presented in this paper will be applied to Bose-Einstein condensates with monopolar 1/r and dipolar 1/r^3 interaction in the subsequent paper [S. Rau et al., Phys. Rev. A, submitted], where we will present a wealth of new phenomena obtained by using the ansatz with coupled Gaussian functions.