We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with a Jacobian elliptic function potential. In the limit of a sinusoidal potential our solutions model a dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided the ratio of the height of the variations of the condensate to its DC offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Numerical simulations suggest such stationary states are experimentally observable.The dilute-gas Bose-Einstein condesate (BEC) in the quasi-one-dimensional regime is modeled by the cubic nonlinear Schrödinger equation (NLS) with a potential [1][2][3]. The various traps which are used to contain the BEC have spurred the solution of the NLS with new potentials [4,5]. BECs trapped in a standing light wave have been used to study phase coherence [6] and matter-wave diffraction [7] and have been predicted to have applications in quantum logic [8] and matter-wave transport [9]. Exact solutions have been obtained for the Kronig-Penney potential [10] and some researchers have used a Bloch function description [11]. In this letter, we study new explicit solutions of the NLS with a Jacobian elliptic function potential.We consider the mean-field model of a quasi-onedimensional repulsive BEC trapped in an external potential which is given by the nonlinear Schrödinger equation [1]In experiments, the trapping potential is generated by a standing light wave [6]. As a model for such a potential we use the periodic potentialwhere sn(x, k) denotes the Jacobian elliptic sine function [12] with elliptic modulus 0 ≤ k ≤ 1. In the limit k → 1 − , V (x) becomes an array of well-separated hyperbolic secant potential barriers or wells, while in the limit k → 0 + it becomes purely sinusoidal. We note that for intermediate values (e.g. k = 1/2) the potential closely resembles the sinusoidal behavior and thus provides a good approximation to the standing wave potential generated experimentally [6]. We present stationary solutions in closed form and study their stability analytically and numerically. We begin by constructing solutions to Eq. (1) which have the formwherewhere B determines a mean amplitude and acts as a DC offset for the number of condensed atoms. The strength of the nonlinearity, which for the BEC is a function of both the atomic coupling and the number of condensed atoms, is determined by the parameters V 0 + k 2 and B, as is apparent in the amplitude of the solutions given by Eq. (3). Note that if x is scaled so that V (x) undergoes only a single oscillation on the ring (in the limit k → 1) the Jacobian elliptic potential provides a model of a single barrier or well [13]. For simplicity we focus on two special cases: (1) k arbitrary and trivial phase (c = 0), and (2) k = 0 with nontrivial phase (c = 0) so that the solutions are trigonometric functions. Trivial Phase Case -In the limit of c = 0, the solutions given in Eqs. (3)-(4) reduce tovalid for V 0 ≥ −k 2 , and ψ(x, t) = −(V 0 +k 2...

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The cubic nonlinear Schrödinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytic and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schrödinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.