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arXiv:1002.1393 · Physical Review A

Two-dimensional gap solitons in elliptic-lattice potentials

Yingji He, Boris A. Malomed, Bambi Hu

Published Mar 8, 2010• 2 citations
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Abstract

We study two-dimensional (2D) matter-wave gap solitons trapped in an elliptically deformed concentric lattice potential, within the framework of the Gross-Pitaevskii equation (GPE) with self-attraction or self-repulsion. For a fixed eccentricity of the lattice, soliton families are found in both the repulsive and attractive models. In the former case, the analysis reveals two kinds of gap solitons trapped in the first oval trough (the ring-shaped potential minimum closest to the center): elliptic annular solitons (EASs), and double solitons (DSs), which are formed by two tightly localized density peaks located at diametrically opposite points of the trough, with zero phase difference between them. With the decrease of the norm, the density distribution in the EAS along the azimuthal direction changes from nearly uniform to double-peaked and, eventually, to the DS. In the attractive model, there exist only DSs in the oval trough, while EASs are not found. All such solitons without the angular momentum ($l=0$) are fully stable. For $l\ensuremath{\ne}0$, vortical solitons---both EASs with a sufficiently large norm (in the repulsive model) and DSs (in models with both signs of the nonlinearity)---are quasistable, exhibiting rocking motion in the elliptic trough (we consider the cases of $l=1 \mathrm{and} l=2$). At smaller values of the norm, the vortical annular solitons (in the repulsive model) are unstable. Stable fundamental solitons trapped in the central potential well are investigated, too, in both the attractive and repulsive models, by means of the variational approximation and numerical methods.

DOI
10.1103/physreva.81.033805

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We investigate two-dimensional (2D) states in Bose-Einstein condensates with self-attraction or self-repulsion, trapped in an axially symmetric optical-lattice potential periodic along the radius. The states trapped both in the central potential well and in remote circular troughs are studied. In the repulsive mode, a new soliton species is found, in the form of radial gap solitons. The latter solitons are completely stable if they carry zero vorticity (l=0) , while with l not equal 0 they develop a weak azimuthal modulation, which makes them rotating patterns, that persist indefinitely long. In addition, annular gap solitons may support stable azimuthal dark-soliton pairs on their crests. In remote troughs of the attractive model, stable localized states may assume a ringlike shape with weak azimuthal modulation, or shrink into solitons strongly localized in the azimuthal direction, which is explained in the framework of an averaged 1D equation with the cyclic azimuthal coordinate. Numerical simulations of the attractive model also reveal stable necklacelike patterns, built of several strongly localized peaks. Dynamics of strongly localized solitons circulating in the troughs is studied too. While the solitons with sufficiently small velocities are completely stable, fast solitons gradually decay, due to the leakage of matter into the adjacent trough, under the action of the centrifugal force. Investigation of head-on collisions between strongly localized solitons traveling in circular troughs shows that collisions between in-phase solitons in a common trough lead to collapse, while pi-out-of-phase solitons bounce many times, but eventually merge into a single one, without collapsing. In-phase solitons colliding in adjacent circular troughs also tend to merge into a single soliton.
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Using the variational approximation and direct simulations in real and imaginary time, we find stable two-dimensional (2D) and 3D solitons in the self-attractive Gross-Pitaevskii equation (GPE) with a potential which is uniform in one direction (z)(z)(z) and periodic in the others (however, the quasi-1D potentials cannot stabilize 3D solitons). The family of solitons includes single- and multiple-peaked ones. The results apply to Bose-Einstein condensates (BEC's) in optical lattices (OL's) and to spatial or spatiotemporal solitons in layered optical media. This is the first prediction of mobile 2D and 3D solitons in BEC's, as they keep mobility along zzz. Head-on collisions of in-phase solitons lead to their fusion into a collapsing pulse. Slow collisions between two multiple-peaked solitons whose main peaks are separated by an intermediate channel end up with their fusion into one single-peaked soliton in the middle channel, $\ensuremath{\simeq}1∕3$ of the original number of atoms being shed off. Stable localized states in the self-repulsive GPE with the low-dimensional OL combined with a parabolic trap are found too. Two such pulses in one channel perform recurrent elastic collisions, periodically featuring sharp interference patterns in the strong-overlap state.
We construct soliton solutions for a self-attractive Bose-Einstein condensate trapped in a rotating optical lattice. The rotation pivot is set at a local maximum of the lattice potential. We demonstrate that fully localized stable solitons, containing $N\ensuremath{\sim}10,000$ atoms, may be supported by the lattice, rotating along with it, if the angular velocity $\ensuremath{\omega}$ is taken below a critical value ${\ensuremath{\omega}}_{c}$ (which is $\ensuremath{\lesssim}10\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}$). A monotonously increasing dependence of ${\ensuremath{\omega}}_{c}$ on NNN is found. In the regime of rapid rotation, the lattice potential is nearly tantamount to the axisymmetric Bessel potential, with a maximum at the center. The latter potential supports fundamental ring-shaped solitons, and also dipole states, which have a bipolar form of two adjacent ring solitons with opposite signs. Stability regions are found for both species (which is the first example of stable azimuthally uniform solitons in a self-focusing model with a radial potential). Unstable solitons of these types evolve into strongly localized nonrotating states. At smaller $\ensuremath{\omega}$, they start to drift slowly, following the rotating lattice. The second critical value, $\ensuremath{\omega}={\ensuremath{\omega}}_{c}^{(2)}$, is found, below which the drifting solitons are destroyed. The rotating lattice supports no stable states in the interval of ${\ensuremath{\omega}}_{c}<\ensuremath{\omega}<{\ensuremath{\omega}}_{c}^{(2)}$.
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