  Superfluidity and superconductivity are remarkable manifestations of quantum
coherence at a macroscopic scale. The dynamics of superfluids has dominated the
study of these systems for decades now, but a comprehensive theoretical
framework is still lacking. We introduce a local extension of the
time-dependent density functional theory to describe the dynamics of fermionic
superfluids. Within this approach one can correctly represent vortex
quantization, generation, and dynamics, the transition from a superfluid to a
normal phase and a number of other large amplitude collective modes which are
beyond the scope of two-fluid hydrodynamics, Ginzburg-Landau and/or
Gross-Pitaevskii approaches. We illustrate the power of this approach by
studying the generation of quantized vortices, vortex rings, vortex
reconnection, and transition from a superfluid to a normal state in real time
for a unitary Fermi gas. We predict the emergence of a new qualitative
phenomenon in superfluid dynamics of gases, the existence of stable
superfluidity when the systems are stirred with velocities significantly
exceeding the nominal Landau critical velocity in these systems.
