# Numann Malik

460 McBryde Hall, Virginia Tech

225 Stanger Street

Blacksburg, VA 24061-1026

Consider the one-dimensional Gross-Pitaevskii (GP) equation subject to non-vanishing boundary conditions, where the dynamics are much richer. Physically (GP) models the propagation of dark pulses in slab waveguides, and Bose-Einstein condensates on a non-zero background. I am interested in the long-time asymptotics of the perturbations for the vacuum state solution and the black soliton; in particular I'm interested in analyzing the low frequency effects in 1D. The motivation arises from the phenomenon that a shelf develops around dark solitons that propagates with speed determined by the background intensity. While the focusing nonlinear Schrodinger equation exhibits the much better understood bright solitons (pulses that decay rapidly at infinity), in the defocusing regime decaying pulses broaden and bright solitons do not exist. Instead solitons can be found as localized dips in intensity that decay off of a continuous wave background. These dark solitons are termed black when the intensity of the dip goes to zero, and gray otherwise, are also associated with a rapid change in phase across the pulse. The experimental observations of dark solitons in both fibre optics and planar waveguides have sparked significant interest in the asymptotic analysis of their propagation.