We study fundamental and vortical solitons in disk-morphed Bose-Einstein condensates (BECs) subject to strong confinement along the axial direction. Starting from the three-dimensional (3D) Gross-Pitaevskii equation (GPE), we proceed to an effective two-dimensional (2D) nonpolynomial Schr\"odinger equation (NPSE) derived by means of the integration over the axial coordinate. Results produced by the latter equation are in very good agreement with those obtained from the full 3D GPE, including cases when the formal 2D equation with the cubic nonlinearity is unreliable. The 2D NPSE is used to predict the density profiles and dynamical stability of repulsive and attractive BECs with zero and finite topological charges in various planar trapping configurations, including the axisymmetric harmonic confinement and one-dimensional periodic potential. In particular, we find a stable dynamical regime that was not reported before, viz., periodic splitting and recombination of trapped vortices with topological charges 2 or 3 in the self-attractive BEC.
We study fundamental and vortical solitons in disk-morphed Bose-Einstein condensates (BECs) subject to strong confinement along the axial direction. Starting from the three-dimensional (3D) Gross-Pitaevskii equation (GPE), we proceed to an effective two-dimensional (2D) nonpolynomial Schr"odinger equation (NPSE) derived by means of the integration over the axial coordinate. Results produced by the latter equation are in very good agreement with those obtained from the full 3D GPE, including cases when the formal 2D equation with the cubic nonlinearity is unreliable. The 2D NPSE is used to predict the density profiles and dynamical stability of repulsive and attractive BECs with zero and finite topological charges in various planar trapping configurations, including the axisymmetric harmonic confinement and one-dimensional periodic potential. In particular, we find a stable dynamical regime that was not reported before, viz., periodic splitting and recombination of trapped vortices with topological charges 2 or 3 in the self-attractive BEC.
We perform accurate investigation of stability of localized vortices in an effectively two-dimensional (``pancake-shaped'') trapped Bose-Einstein condensate with negative scattering length. The analysis combines computation of the stability eigenvalues and direct simulations. The states with vorticity S=1 are stable in a third of their existence region, 0<N<(1∕3)Nmax(S=1), where N is the number of atoms, and Nmax(S=1) is the corresponding collapse threshold. Stable vortices easily self-trap from arbitrary initial configurations with embedded vorticity. In an adjacent interval, (1∕3)Nmax(S=1)<N<0.43Nmax(S=1), the unstable vortex periodically splits in twofragments and recombines. At N>0.43Nmax(S=1), the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full three-dimensional (3D) Gross-Pitaevskii equation. In a moderately anisotropic 3D configuration, with the aspect ratio 10, the stability interval of the S=1 vortices occupies $\ensuremath{\approx}40%$ of their existence region, hence the two-dimensional (2D) limit provides for a reasonable approximation in this case. For the isotropic 3D configuration, the stability interval expands to 65% of the existence domain. Overall, the vorticity heightens the actual collapse threshold by a factor of up to 2. All vortices with $S\ensuremath{\geqslant}2$ are unstable.
We construct exact solutions of the Gross-Pitaevskii equation for solitary vortices, and approximate ones for fundamental solitons, in two-dimensional 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 Schr\"odinger equation. The VS families in the system with the attractive and repulsive nonlinearity are mutually complementary. Stable VSs with vorticity $S\ensuremath{\geqslant}2$ and those corresponding to higher-order radial states are reported, in the case of the attraction and repulsion, respectively.
Starting from the three-dimensional (3D) Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D nonpolynomial Schr\"odinger equation (2D NPSE) that models disk-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities.
Abstract Starting from the standard three-dimensional (3D) Gross–Pitaevskii equation (GPE) and using a variational approximation, we derive an effective one-dimensional nonpolynomial Schrödinger equation (1D-NPSE) governing the axial dynamics of atomic Bose–Einstein condensates (BECs) under the action of a singular but physically relevant funnel-shaped transverse trap, i.e. an attractive 2D potential ∼−1/ r (where r is the radial coordinate in the transverse plane), in combination with the repulsive self-interaction. Wave functions of the trapped BEC are regular, in spite of the potential’s singularity. The model applies to a condensate of particles (small molecules) carrying a permanent electric dipole moment in the field of a uniformly charged axial thread, as well as to a quantum gas of magnetic atoms pulled by an axial electric current. By means of numerical simulations, we verify that the effective 1D-NPSE provides accurate static and dynamical results, in comparison to the full 3D GPE, for both repulsive and attractive signs of the intrinsic nonlinearity.
We consider a two-dimensional (2D) counterpart of the experiment that led to the creation of quasi-1D bright solitons in Bose-Einstein condensates (BECs) [Nature 417, 150--153 (2002)]. We start by identifying the ground state of the 2D Gross-Pitaevskii equation for repulsive interactions, with a harmonic-oscillator (HO) trap, and with or without an optical lattice (OL). Subsequently, we switch the sign of the interaction to induce interatomic attraction and monitor the ensuing dynamics. Regions of the stable self-trapping and catastrophic collapse of 2D fundamental solitons are identified in the parameter plane of the OL strength and BEC norm. The increase of the OL strength expands the persistence domain for the solitons to larger norms. For single-charged solitary vortices, in addition to the survival and collapse regimes, an intermediate one is identified, where the vortex resists the collapse but loses its structure, transforming into a fundamental soliton. The same setting may also be implemented in the context of optical solitons and vortices, using photonic-crystal fibers.
We consider, both analytically and numerically, the evolution of two-dimensional (2D) nonlinear matter-wave pulses in a Bose-Einstein condensate with a disk-shaped trap and repulsive atom-atom interactions. Due to the strong confinement in the axial direction the sound speed of the system is c=(1/21/4)c0, where c0 is the corresponding value without the trap. From the 3D order-parameter equation of the condensate, we derive a soliton-bearing Kadomtsev-Petriashvili equation with positive dispersion. When the trapping potential is weak in two transverse directions, a low-depth plane dark soliton can propagate in the condensate with a changing profile but preserving its structure down to the boundary of the condensate. We show that high-depth plane dark solitons are unstable to long-wavelength transverse disturbances. The instability appears as a longitudinal modulation of the soliton amplitude decaying into vortices. We also show how a dark lumplike 2D nonlinear excitation can be excited in the system. Furthermore, a dark lump decaying algebraically in two spatial directions can propagate rather stable in the condensate, but disappears near the boundary of the condensate where two vortices are nucleated. The vortices move in opposite directions along the boundary and when meeting merge creating a new lump. Finally, we also provide results for head-on and oblique collisions of two lumps in the system.