A Hermite pseudo-spectral method for solving systems of Gross-Pitaevskii equations. (English) Zbl 1151.35087

Weishäuptl, Rada M.; Schmeiser, Christian; Markowich, Peter A.; Borgna, Juan Pablo

Commun. Math. Sci. 5, No. 2, 299-312 (2007).

It is known that the full three-dimensional Gross-Pitaevskii equation, which describes the behavior of Bose-Einstein condensates in the mean-field approximation, can be reduced to a low-dimensional (two- or one-dimensional) equation of the same type, assuming that the condensate is subjected to strong confinement in one or two transverse directions, respectively. Usually, the low-dimensional equation is derived by assuming that the three-dimensional wave function is approximately factorised into a ground state of the respective strongly confining transverse potential, multiplied by an arbitrary finction of time and the remaining unconfined coordinate(s).
This paper addresses a situation when the confinement is not very strong, hence one should approximate the three-dimensional wave function by a combination of several terms, each being a product of the ground state (\(k=0\)) or one of excited states (\(k\geq 1\)) of the confining potential, times respective arbitrary functions \(\varphi_k\) of time and remaining coordinate(s), \(x\). The so derived truncated system of equations takes the following general form:
\[ i\frac{\partial\varphi_k}{\partial t} = H_{\perp}\varphi_k + \sum \gamma_{klmn} \varphi_l\varphi_n\varphi_m^{*}, \]
where \( H_{\perp}\) is the Hamiltonian acting on coordinates \(x\) in the linear approximation, the asterisk stands for the complex conjugations, and coupling coefficients \(\gamma_{klmn}\) are determined by overlapping integrals of respective eigenstates of the confining potential. The paper elaborates a method for numerical simulations of this coupled nonlinear system (in fact, for the system of four equations). The method is based on a pseudo-spectral Hermite-expansion technique, combined with a Crank-Nicolson scheme. Numerical experiments are presented for the case when the remaining unconfined space is one-dimensional (which corresponds to the “cigar-shaped” trap), and results are compared to those obtained by means of an alternative method.

Reviewer: Boris A. Malomed (Tel Aviv)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs

Keywords:

Gross-Pitaevskii equation; Hermite polynomial; harmonic oscillator; Crank-Nicolson scheme