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arXiv:1302.0445 · Modern Physics Letters B

EXACT SOLUTIONS TO THE THREE-DIMENSIONAL GROSS–PITAEVSKII EQUATION WITH MODULATED RADIAL NONLINEARITY

WEI-TING WANG, YING-YING LI, SHI-JIE YANG

Published Apr 10, 2013• 0 citations
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Abstract

We study the Bose–Einstein condensate trapped in a three-dimensional spherically symmetrical potential. Exact solutions to the stationary Gross–Pitaevskii equation are obtained for properly modulated radial nonlinearity. The solutions contain vortices with different winding numbers and exhibit the shell-soliton feature in the radial distributions.

DOI
10.1142/s0217984913501054

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We study the Bose–Einstein condensate trapped in a three-dimensional spherically symmetrical potential. Exact solutions to the stationary Gross–Pitaevskii equation are obtained for properly modulated radial nonlinearity. The solutions contain vortices with different winding numbers and exhibit the shell-soliton feature in the radial distributions.
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We construct exact solutions of the Gross-Pitaevskii equation for solitary vortices, and approximate ones for fundamental solitons, in 2D models of Bose-Einstein condensates with a spatially modulated nonlinearity of either sign and a harmonic trapping potential. The number of vortex-soliton (VS) modes is determined by the discrete energy spectrum of a related linear Schrödinger equation. The VS families in the system with the attractive and repulsive nonlinearity are mutually complementary. \emph{% Stable} VSs with vorticity S≥2S\geq 2S≥2 and those corresponding to higher-order radial states are reported for the first time, in the case of the attraction and repulsion, respectively.
Exact solitonic solutions of the Gross–Pitaevskii equation with time-modulated nonlinearity of a(t) = a0 / (t + t0) are obtained. With help of these solutions, we analyze the properties of Feshbach-managed solitons in Bose–Einstein condensates in details. Our results show that the parameters of atomic matter waves can be manipulated by proper variation of the scattering length. In particular, an exact two-soliton solution is given, from which, it is shown that the separation between the neighboring solitons can be effectively maintained by allowing the solitons to have unequal initial amplitudes.
The form and stability properties of axisymmetric and spherically symmetric stationary states in two and three dimensions, respectively, are elucidated for Bose-Einstein condensates. These states include the ground state, central vortices, and radial excitations of both. The latter are called ring solitons in two dimensions and spherical shells in three. The nonlinear Schr\"odinger equation is taken as the fundamental model; both extended and harmonically trapped condensates are considered. It is found that the instability times of ring solitons can be long compared to experimental time scales, making them effectively stable over the lifetime of an experiment.
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