Stationary states of the Gross-Pitaevskii equation with linear counterpart

Details Stationary states of the Gross-Pitaevskii equation with linear counterpart Stationary states of the Gross-Pitaevskii equation with linear counterpart

1998-09-14

arxiv.org/abs/cond-mat/9809203v3

Phys. Lett. A 275, 424-434 (2000)

Physics

Condensed Matter

15 pages, 6 figures, LaTeX

Scientific paper

10.1016/S0375-9601(00)00619-8

We study the stationary solutions of the Gross-Pitaevskii equation that reduce, in the limit of vanishing non-linearity, to the eigenfunctions of the associated Schr\"odinger equation. By providing analytical and numerical support, we conjecture an existence condition for these solutions in terms of the ratio between their proper frequency (chemical potential) and the corresponding linear eigenvalue. We also give approximate expressions for the stationary solutions which become exact in the opposite limit of strong non-linearity. For one-dimensional systems these solutions have the form of a chain of dark or bright solitons depending on the sign of the non-linearity. We demonstrate that in the case of negative non-linearity (attractive interaction) the norm of the solutions is always bounded for dimensions greater than one.

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