Localized matter-wave patterns with attractive interaction in rotating potentials

We consider a two-dimensional (2D) model of a rotating attractive Bose-Einstein condensate (BEC), trapped in an external potential. First, a harmonic potential with the critical strength is considered, which generates quasisolitons at the lowest Landau level (LLL). We describe a family of the LLL quasisolitons using both numerical method and a variational approximation (VA), which are in good agreement with each other. We demonstrate that kicking the LLL mode or applying a ramp potential sets it in the Larmor (cyclotron) motion that can also be accurately modeled by the VA. Collisions between two such moving modes may be elastic or inelastic depending on their total norm. If an additional confining potential is applied along with the ramp, it creates a stationary edge state. Applying a kick to the edge state in the direction of the ramp gives rise to a skipping motion in the perpendicular direction. These regimes may be interpreted as the Hall effect for the quasisolitons. Next, we consider the condensate trapped in an axisymmetric quartic potential. Three species of localized states and their stability regions are identified, viz., vortices with arbitrary topological charge m, "crescents" (mixed-vorticity states), and strongly localized center-of-mass (c.m.) states, alias quasisolitons, shifted off the rotation pivot. These results are similar to those reported before for the model with a combined quadratic-quartic trap. Stable pairs of c.m. states set at diametrically opposite points are found, too. We present a VA which provides for an accurate description of vortices with all values of m, and of the c.m. states. We also demonstrate that kicking them in the azimuthal direction sets the quasisolitons in epitrochoidal motion (which is also accurately predicted by the VA), collisions between them being elastic.