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arXiv:cond-mat/0007117 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Stability of stationary states in the cubic nonlinear Schrödinger equation: Applications to the Bose-Einstein condensate

Lincoln D. Carr, J. Nathan Kutz, William P. Reinhardt

Published May 18, 2001• 67 citations
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Abstract

The cubic nonlinear Schrödinger equation is the quasi-one-dimensional limit of the mean-field theory which models dilute gas Bose-Einstein condensates. Stationary solutions of this equation can be characterized as soliton trains. It is demonstrated that for repulsive nonlinearity a soliton train is stable to initial stochastic perturbation, while for attractive nonlinearity its behavior depends on the spacing between individual solitons in the train. Toroidal and harmonic confinement, both of experimental interest for Bose-Einstein condensates, are considered.

DOI
10.1103/physreve.63.066604

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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 analytical 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.
The cubic nonlinear Schrodinger equation with repulsive nonlinearity and elliptic function potential in two-dimensions models a repulsive dilute gas Bose--Einstein condensate in a lattice potential. A family of exact stationary solutions is presented and its stability is examined using analytical and numerical methods. All stable trivial-phase solutions are off-set from the zero level. Our results imply that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Key words: Bose-Einstein condensates, two-dimensional nonlinear Schrodinger equation, periodic potential, elliptic functions PACS: 65.65 1 Introduction The cubic nonlinear Schrodinger equation with attractive or repulsive nonlinearity and a potential is used as a mean-field model for the dynamics of a dilute-gas Bose Einstein condensate (BEC) [1,2]. In this case, the equation is often referred to as the Gross-Pitaevskii equation. These BECs are of interest in both the theoretical and experim...
Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schr\"odinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger equation. Their stability is examined using analytic and numerical methods. Trivial-phase solutions are experimentally stable provided they have nodes and their density is localized in the troughs of the potential. Stable time-periodic solutions are also examined.
We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with an elliptic function potential. In the limit of a sinusoidal potential our solutions model a quasi-one-dimensional dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided that 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. Recent developments allow for experimental investigation of these predictions.
It is shown that the quasi-one-dimensional Bose-Einstein condensate is experimentally accessible and rich in intriguing phenomena. We demonstrate numerically and analytically the existence, stability and perturbation-induced dynamics of all types of stationary states of the quasi-one-dimensional nonlinear Schrödinger equation for both repulsive and attractive cases. Among our results are: the connection between stationary states and solitons; creation of vortices from such states; manipulation of such states with simple phase profiles; demonstration of the fragility of the condensate phase in response to shock; and a robust stabilization of the attractive Bose-Einstein condensate.