Physics Letters A 377 (2013) 2408–2415 Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla A variational approach to the modulational-oscillatory instability of Bose–Einstein condensates in an optical potential S. Sabari a , E. Wamba b , K. Porsezian a,∗ , A. Mohamadou c,d , T.C. Kofané b a Department of Physics, Pondicherry University, Puducherry 605014, India b Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon c Condensed Matter Laboratory, Department of Physics, Faculty of Science, University of Douala, P.O. Box 24157, Douala, Cameroon d The Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, Strada Costiera 11, I-34014 Trieste, Italy article info abstract Article history: Received 11 March 2013 Received in revised form 14 June 2013 Accepted 1 July 2013 Available online 5 July 2013 Communicated by A.R. Bishop Keywords: Variational approach Gross–Pitaevskii equation Optical potential Modulational instability Oscillatory instability We use the time-dependent variational approach to demonstrate how the modulational and oscillatory instabilities can be generated in Bose–Einstein condensates (BECs) trapped in a periodic optical lattice with weak driving harmonic potential. We derive and analyze the ordinary differential equations for the time evolution of the amplitude and phase of the modulational perturbation, and obtain the instability condition of the condensates through the effective potential. The effect of the optical potential on the dynamics of the BECs is shown. We perform direct numerical simulations to support our theoretical findings, and good agreement is found.  2013 Elsevier B.V. All rights reserved. 1. Introduction Inspired by the work of Bose and Einstein in 1925, the first experimental demonstration of Bose–Einstein condensates (BECs) was carried out with dilute alkali gases recently in 1995 [1–3]. This achievement has then triggered numerous research works in the field of ultracold atoms. Among them the experimental [4–6] and theoretical [7–17] investigations of the dynamics and properties of BECs in optical lattice (OL) potentials have been a central topic. In these experiments, the OL is created by two counterpropagating laser beams forming a standing wave interference pattern. The dy- namic properties of the atoms are characterized by the depth and the period of this optically-induced potential. The intensity of the OL potential can be modulated from very weak to very strong [4]. Hence, BECs in periodic potentials have been found useful in inves- tigating many physical phenomena such as Josephson effect [18], Bloch oscillations [4,19,20], Landau–Zener tunneling [21], solitons [22], quantum phase transitions of the Mott insulator type [23], superfluid and dissipative dynamics [24], phase diagram [25], and nonlinear dynamics of a dipolar [26] or spinor [27] BEC. The theoretical model that gives a satisfactory description of the dynamics of BECs is the mean-field Gross–Pitaevskii (GP) equa- tion [28]. Nonlinear terms arise in the GP equation to account for * Corresponding author. Tel.: +91 413 2654403; fax: +91 413 2655183. E-mail address: ponzsol@gmail.com (K. Porsezian). the effect of interatomic interactions in the condensate. One of the most attractive features of cold atomic gases is how the in- teratomic interaction affects the nonlinear dynamical properties of condensates. Two-body interatomic interactions in BECs are mod- eled through the s-wave scattering length, a s , which may be either negative or positive, meaning that the interaction is attractive or repulsive, respectively [28]. The strength and sign of the atomic scattering length can be varied by tuning the external magnetic field near Feshbach resonance [29]. This indicates that the inter- action strength can be controlled by using different experimental devices. As is well known, the repulsive BECs in OL can give rise to stable localized matter-wave states in the form of gap soli- tons. These BEC gap solitons were predicted theoretically [30–32] and demonstrated experimentally [6]. Gap solitons are represented by stationary solutions to the respective GP equation, with the eigenvalue (chemical potential) located in a finite bandgap of the OL-induced spectrum [33]. In the BECs with attractive interactions (a s < 0), solitons have been realized in the ground state of the con- densate. Such solitons were created in condensates of 7 Li [34] and 85 Rb [35] atoms, with the sign of the atomic interactions switched from negative to positive by means of the Feshbach-resonance technique (in the latter case, the solitons were observed in a post- collapse state of the condensate). In the presence of a periodic potential, such solitons should exist too, with the chemical poten- tial falling in the semi-infinite gap of the spectrum, as first shown in the context of the optical setting [8], and later demonstrated in details in the framework of the GP equation [36,37]. 0375-9601/$ – see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.07.005