Filamentary Dynamics of Dark Solitons, Vortex Rings and Knots*

In the present talk, we will revisit some principal excitations in self-repulsive Bose-Einstein condensates, namely dark solitons, vortex rings and knots. For dark solitons, upon introducing them and explaining their existence and stability properties in 1d, we will extend them both in the form of stripes and in that of rings in two-dimensions, presenting an alternative (adiabatic-invariant based) formulation of their stability and excitations. We will explore their filamentary dynamics, as well as the states that emerge from their transverse (snaking) instability. Then, we will consider these structures even in three dimensions, in the form of planar, as well as spherical shell solitons and generalize our adiabatic invariant formulation there. We will show how the formalism can be extended to the case of vortex rings and their own filamentary dynamics.
Finally, time permitting, we will give some glimpses of how some of these dynamical features in 1d and 2d generalize in a multi-orbital, time-dependent quantum setting for dark solitons, as well as mention about the dynamics of knots in 3d BECs.