  Ultrashort optical pulses with orbital angular momentum (OAM) owing to their
various applications in classical as well as quantum domains attract a lot of
research attention. The evolution of their spatio-temporal dynamics is of
particular interest. In the present work, we numerically explore the
spatio-temporal evolution of ultrashort Laguerre Gauss (LG) vortex with power
exponential spectrum, also called power exponential (PE) pulse, in dispersive
and nonlinear media. We in particular address the evolution of space-time
non-separable ultrashort vortex pulse. The study reveals the asymmetric
spatio-temporal evolution and splitting in the spatio-temporal plane. We
observe compression at the trailing edge of the pulse with increase in the
strength of the nonlinearity. The study further reveals that the temporal,
spectral and, chirp evolution patterns of the pulse vary uniquely across
different spatial regions and this variation depends upon the strength of
nonlinearity and propagation distance. In weak nonlinear regime, a redshift in
spectrum is noted as we move radially outward from the bright caustic of
ultrashort LG vortex while a blue shifted spectrum emerges on approaching
towards the phase singularity. Interestingly, as the pulse furthers in the
medium, nonlinearity induced trends in spectral shift is witnessed. With
stronger nonlinearity, a complex spectral evolution is observed. Furthermore,
we study the evolution of frequency chirp with propagation distance both with
weak and strong nonlinearities.
