Physics Letters A 354 (2006) 115–118 www.elsevier.com/locate/pla Bose–Einstein condensates: Analytical methods for the Gross–Pitaevskii equation Carlos Trallero-Giner a , J. Drake a , V. Lopez-Richard b,∗ , C. Trallero-Herrero c , Joseph L. Birman d a Faculty of Physics, Havana University, 10400 Havana, Cuba b F.F.C.L.RP, Departamento de Física e Matemática, Universidade de São Paulo, 14040-901 Ribeirão Preto, SP, Brazil c Physics and Astronomy Department, SUNY at Stony Brook, NY 11794-3800, USA d Department of Physics, The City College of CUNY, New York, NY 10031, USA Received 30 June 2005; received in revised form 4 January 2006; accepted 11 January 2006 Available online 23 January 2006 Communicated by A.R. Bishop Abstract We present simple analytical methods for solving the Gross–Pitaevskii equation (GPE) for the Bose–Einstein condensation (BEC) in the dilute atomic alkali gases. Using a soliton variational Ansatz we consider the effects of repulsive and attractive effective nonlinear interactions on the BEC ground state. We perform a comparative analysis of the solutions obtained by the variational Ansatz, the perturbation theory, the Thomas– Fermi approximation, and the Green function method with the numerical solution of the GPE finding universal ranges where these solutions can be used to predict properties of the condensates. Also, a generalization of the soliton variational approach for two-species of alkali atoms of the GPE is performed as a function of the effective interaction λ i (i = 1, 2) and the inter-species λ 12 and λ 21 constants.  2006 Elsevier B.V. All rights reserved. Bose–Einstein condensates (BEC) have been reported in Li, Rb, and Na atoms [1]. In most cases the weakly interacting boson of the Gross–Pitaevskii (GPE) theory has been invoked to describe the properties of the condensate [2,3]. Nowadays, one of the most spectacular research is devoted to discuss the dynamics of BEC and the appearance of solitons based upon the nonlinear Schrödinger equation. The first bright soliton was created for negative scattering length BEC [4,5] and, for re- pulsive interaction, dark solitons have also been observed [6]. Recently, Eiermann et al. [7] reported experimental observation of different type of bright matter wave solitons for 87 Rb atoms in a periodic potential, allowing to study the dynamics of the condensate in a quasi-one-dimensional waveguide. Most of the theoretical work has been devoted to implement numerical so- lutions of the GP equation for the order parameter (see [3] and references therein). To design and to control the parameters of the condensate it is very useful to get analytical expressions for the chemical potential and for the order parameter as well. * Corresponding author. E-mail address: vlopez@df.ufscar.br (V. Lopez-Richard). Our goal is to describe the dynamics of the dilute ultra-cold atom cloud in the BE condensed phase, implementing new an- alytical approaches beyond the Thomas–Fermi (TF) limit and perturbation theory [8]. In order to get analytical expressions, several authors have used a Gaussian as a trial wave function since in the linear limit one can obtain the linear Schrödinger equation in a parabolic potential [9]. For the negative scatter- ing length case, a Gaussian function should probably be a good Ansatz to describe the condensate. Nevertheless, for repulsive atom–atom interaction, the shape of the wave function is simi- lar to the TF solution (at least in the strong nonlinear limit). As it is well known, the validity of the variational result is in many cases qualitative and even if the shape of the order parameter is near to the trial wave function, this method is not always a good procedure for solving a nonlinear equation. In this Let- ter we present an efficient algorithm for the calculation of the ground state of the nonlinear Schrödinger equation (NLSE) in a harmonic trap using the Green function formalism and also a “soliton” variational solution. For sake of simplicity and in or- der to validate methodologically the obtained solutions, we just consider the isomorphic one-dimensional nonlinear GPE [2,3], 0375-9601/$ – see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.032