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arXiv:0801.3307 · Physical Review A

Solitons in two-dimensional Bose-Einstein condensates

Shunji Tsuchiya, F. Dalfovo, Лев П. Питаевский

Published Apr 3, 2008• 50 citations
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Abstract

The excitations of a two-dimensional (2D) Bose-Einstein condensate in the presence of a soliton are studied by solving the Kadomtsev-Petviashvili equation which is valid when the velocity of the soliton approaches the speed of sound. The excitation spectrum is found to contain states which are localized near the soliton and have a dispersion law similar to the one of the stable branch of transverse oscillations of a 1D gray soliton in a 2D condensate. By using the stabilization method we show that these localized excitations behave as resonant states coupled to the continuum of free excitations of the condensate.

DOI
10.1103/physreva.77.045601

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The excitations of a two-dimensional (2D) Bose-Einstein condensate in the presence of a soliton are studied by solving the Kadomtsev-Petviashvili equation which is valid when the velocity of the soliton approaches the speed of sound. The excitation spectrum is found to contain states which are localized near the soliton and have a dispersion law similar to the one of the stable branch of transverse oscillations of a 1D gray soliton in a 2D condensate. By using the stabilization method we show that these localized excitations behave as resonant states coupled to the continuum of free excitations of the condensate.
For two-component disk-shaped Bose–Einstein condensates with repulsive atom-atom interaction, the small amplitude, finite and long wavelength nonlinear waves can be described by a Kadomtsev–Petviashvili-I equation at the lowest order from the original coupled Gross–Pitaevskii equations. One- and two-soliton solutions of the Kadomtsev–Petviashvili-I equation are given, therefore, the wave functions of both atomic gases are obtained as well. The instability of a soliton under higher-order long wavelength disturbance has been investigated. It is found that the instability depends on the angle between two directions of both soliton and disturbance.
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,{c=(1/2}^{1/4}{)c}_{0},c=(1/21/4)c0​, where c0{c}_{0}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.
We analyze the dynamical stability of gap solitons formed in a quasi-one-dimensional Bose-Einstein condensate in an optical lattice. Using two different numerical methods we show that, under realistic assumptions for experimental parameters, a gap soliton is stable only in a truly one-dimensional situation. In two and three dimensions, resonant transverse excitations lead to dynamical instability. The time scale of the decay is numerically calculated and shown to be large compared to the characteristic time scale of solitons for realistic physical parameters.
For the first time, we find the complex solitons for a quasi-one-dimensional Bose-Einstein condensate with two-and three-body interactions. These localized solutions are characterized by a power law behaviour. Both dark and right solitons can be excited in the experimentally allowed parameter domain, when two-and three-body interactions are,respectively, repulsive and attractive. The dark solitons travel with a constant speed, which is quite different from the Lieb mode, where profiles with different speeds, bounded above by sound velocity, can exist for specified interaction strengths. We also study the properties of these solitons in the presence of harmonic confinement with time-dependent nonlinearity and loss. The modulational instability and the Vakhitov-Kolokolov criterion of stability are also studied.