Modulated amplitude waves in collisionally inhomogeneous Bose–Einstein condensates

Abstract

We investigate the dynamics of an effectively one-dimensional Bose–Einstein condensate (BEC) with scattering length a subjected to a spatially periodic modulation, a=a(x)=a(x+L). This "collisionally inhomogeneous" BEC is described by a Gross–Pitaevskii (GP) equation whose nonlinearity coefficient is a periodic function of x. We transform this equation into a GP equation with a constant coefficient and an additional effective potential and study a class of extended wave solutions of the transformed equation. For weak underlying inhomogeneity, the effective potential takes a form resembling a superlattice, and the amplitude dynamics of the solutions of the constant-coefficient GP equation obey a nonlinear generalization of the Ince equation. In the small-amplitude limit, we use averaging to construct analytical solutions for modulated amplitude waves (MAWs), whose stability we subsequently examine using both numerical simulations of the original GP equation and fixed-point computations with the MAWs as numerically exact solutions. We show that "on-site" solutions, whose maxima correspond to maxima of a(x), are more robust and likely to be observed than their "off-site" counterparts.

Additional Information

Author preprint via arXiv -- arXiv:nlin/0607009v2 Published version, Copyright © 2007 Elsevier B.V. Received 7 July 2006; revised 23 December 2006; accepted 14 February 2007. Communicated by S. Kai. Available online 13 March 2007. We thank Jared Bronski and Richard Rand for useful comments during the preparation of this paper. M.A.P. acknowledges support from the Gordon and Betty Moore Foundation through Caltech's Center for the Physics of Information. P.G.K. acknowledges funding from NSF-DMS-0204585, NSF-CAREER, and NSF-DMS-0505663. B.A.M. appreciates partial support from the Israel Science Foundation through the Excellence-Center grant No. 8006/03. D.J.F. acknowledges partial support from the Special Research Account of the University of Athens.

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