  In this work, we numerically consider the initial value problem for nonlinear
Schr\"odinger (NLS) type models arising in the physics of ultracold boson
gases, with generic Gaussian wavepacket initial data. The corresponding
Gaussian's width and, wherever relevant also its amplitude, serve as control
parameters. First we explore the one-dimensional, standard NLS equation with
general power law nonlinearity, in which large amplitude excitations
reminiscent of Peregrine solitons or regular solitons appear to form, as the
width of the relevant Gaussian is varied. Furthermore, the variation of the
nonlinearity exponent aims at a first glimpse of the interplay between rogue or
soliton formation and collapse features. The robustness of the main features to
noise in the initial data is also confirmed. To better connect our study with
the physics of atomic condensates, and explore the role of dimensionality
effects, we also consider the nonpolynomial Schr\"odinger equation (NPSE), as
well as the full three-dimensional NLS equation, and examine the degree to
which relevant considerations generalize.
