We investigate the effects of vortex interaction on the formation of interference patterns in a coherent pair of two-dimensional Bose condensed clouds of ultra-cold atoms traveling in opposite directions subject to a harmonic trapping potential. We i
dentify linear and nonlinear regimes in the dipole oscillations of the condensates according to the balance of internal and centre-of-mass energies of the clouds. Simulations of the collision of two clouds each containing a vortex with different winding number (charge) were carried out in these regimes in order to investigate the creation of varying interference patterns. The interaction between different vortex type can be clearly distinguished by those patterns.
The quasiparticle excitation spectra of a Bose gas trapped in a highly anisotropic trap is studied with respect to varying total number of particles by numerically solving the effective one-dimensional (1D) Gross-Pitaevskii (GP) equation proposed rec
ently by Mateo textit{et al.}. We obtain the static properties and Bogoliubov spectra of the system in the high energy domain. This method is computationally efficient and highly accurate for a condensate system undergoing a 1D to three-dimensional (3D) cigar-shaped transition, as is shown through a comparison our results with both those calculated by the 3D-GP equation and analytical results obtained in limiting cases. We identify the applicable parameter space for the effective 1D-GP equation and find that this equation fails to describe a system with large number of atoms. We also identify that the description of the transition from 1D Bose-Einstein condensate (BEC) to 3D cigar-shaped BEC using this equation is not smooth, which highlights the fact that for a finite value of $a_perp/a_s$ the junction between the 1D and 3D crossover is not perfect.
