  In this work, we present a detailed study of the dynamics and stability of
fundamental spatiotemporal solitons emerging in multimode waveguides with a
parabolic transverse profile of the linear refractive index. Pulsed beam
propagation in these structures can be described by using a Gross-Pitaevskii
equation with a two-dimensional parabolic spatial potential. Our investigations
are based on comparing variational approaches, based on the Ritz optimization
method, with extensive numerical simulations. We found that, with a Kerr
self-focusing nonlinearity, spatiotemporal solitons are stable for low pulse
energies, where our analytical results find a perfect agreement with the
numerical simulations. However, solitons with progressively increasing energies
eventually undergo wave collapse, which is not predicted within the variational
framework. In a self-defocusing scenario, again for low energies there is good
agreement between the variational predictions and simulations. Whereas, for
large soliton energies complex spatiotemporal dynamics emerge.
