Dynamics of overlapping vortices in complex scalar fields

We investigate dynamics of overlapping vortices in the nonlinear Schr\"{o}dinger equation, the nonlinear heat equation and in the equation with an intermediate Schr\"{o}dinger-diffusion dynamics. Because of formal similarity on a perturbative level we discuss also the nonlinear wave equation (Goldstone model). Special solutions are found like vortex helices, double-helices and braids, breather states and vortex mouths. A pair of vortices in the Goldstone model scatters by the right angle in the head-on collision. It is found that in a dissipative system there is a characteristic lenght scale above which vortices can be entangled but below which the entanglement is unstable.Comment: detailed analysis of the continuation in the complex time plane; latest revision: the paper is made more readable. 9 pages in Late

Images of a Bose-Einstein condensate in position and momentum space

In the Bogoliubov theory a condensate initially prepared in its ground state described by stationary Bogoliubov vacuum and later perturbed by a time-dependent potential or interaction strength evolves into a time-dependent excited state which is dynamical Bogoliubov vacuum. The dynamical vacuum has a simple diagonal form in a time-dependent orthonormal basis of single particle modes. This diagonal representation leads to a gaussian probability distribution for possible outcomes of density measurements in position and momentum space. In these notes we also discuss relations with the U(1) symmetry breaking version of the Bogoliubov theory and give two equivalent gaussian integral representations of the dynamical vacuum state.Comment: 4 pages; Talk given at the Laser Physics Workshop, July 2005, Kyoto, Japa