Exact stationary wave patterns in three coupled nonlinear Schrödinger/Gross–Pitaevskii equations Zhenya Yan a,b, * , K.W. Chow c , Boris A. Malomed c,d a Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100080, China b International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China c Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong d Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69987, Israel article info Article history: Accepted 3 April 2009 Communicated by: Prof. M. Wadati abstract The evolution of a Bose–Einstein condensate (BEC) with an internal degree of freedom, i.e., spinor BEC, is governed by a system of three coupled mean-field equations. The system admits the application of the inverse scattering transform and Hirota bilinear method under appropriate conditions, which makes it possible to generate exact analytical solutions relevant to physical applications. Here, we produce six families of exact periodic solutions, directly constructed in terms of Jacobi elliptic functions. Solitary-wave limit forms, obtained from these solutions in the long-wave limit, are presented too. Ó 2009 Published by Elsevier Ltd. 1. Introduction The dynamics of Bose–Einstein condensates (BECs) is a subject of great interest to experimentalists and theorists alike. The mean-field theoretical description of BEC is based on the Gross–Pitaevskii, alias nonlinear Schrödinger, equations with cubic nonlinear terms, which, in particular, correctly predict matter-wave solitons in various configurations of condensates with attractive and repulsive interactions (see recent collection of reviews [1]). Especially rich is the multi-component dynamics of spinor BECs. In the presence of an optical trap, one-dimensional spi- nor condensates exhibit ‘ferromagnetic’ (single-component) and ‘polar’ (three-component) stationary states, whose stability depends on the scattering lengths of atomic collisions in different angular-momentum channels [2–4]. A model of such a three-component spinor BEC, which permits significant analytical advances, was proposed in [5,6]. This model permits bright solitons with the spin degree of freedom. In particular, for the attractive mean–field nonlinearity and ferromagnetic spin-exchange interaction, the inverse scattering method applies under special constraints, giving rise to multi-soliton solu- tions [7]. On the other hand, when the interactions are repulsive, the respective system of three coupled Gross–Pitaevskii equations (GPEs) admits dark solitons. One- and two-soliton solutions of the dark type can be obtained analytically [8]. Ferromagnetic states feature wave functions shaped as domain walls with a nonzero total spin, while polar states are arranged as hole sol- itons with zero spin, see also [9]. In addition, bound complexes of dark and bright solitons can be found in integrable and nonintegrable versions of the spinor-BEC models [10,11]. The goal of the present work is to obtain novel exact solutions for periodic (cnoidal) wave patterns in the spinor system. We stress that the system of the coupled GPEs, which describes this system, is different from the well-known Manakov’s type, and hence solutions obtained for the latter system [12–16] are not applicable in the present case. 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2009.04.043 * Corresponding author. Address: Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100080, China. E-mail address: zyyan_math@yahoo.com (Z. Yan). Chaos, Solitons and Fractals 42 (2009) 3013–3019 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos